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Investigating field burial by magnetically confined accretion mounds on Neutron Stars

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Mound spreading buries neutron star dipole fields by up to 37 percent

desk verdict Useful methods paper with a genuinely new current-free BC; the 37% burial headline is real but resolution-bound, so treat it as evidence of onset, not a converged number. read the letter →

arxiv 2507.08509 v1 pith:ZIC2E7IG submitted 2025-07-11 astro-ph.HE

Saurabh Yeole , Dipanjan Mukherjee , Ankush Mandal This is my paper · ORCID
classification astro-ph.HE
keywords neutronstarsaccretionmoundsmagneticfieldburialGrad-Shafranovequationmagnetostaticequilibriacurrent-freeboundaryconditionmultipolarfieldsmassellipticity
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 claims that magnetically confined accretion mounds on neutron stars do not stay where they are placed: as they grow, ring-shaped mounds spread latitudinally toward the equator, and this spreading is what buries the star's dipole field. Solving the Grad-Shafranov magnetostatic equations at high resolution with a new current-free outer boundary condition, the authors find that a $10^9$ G neutron star with a mound loaded near the Alfvén truncation radius reduces its normalized dipole moment at the outer radius to $0.627$, a 37 percent reduction, with matter reaching the equator. Previous solutions stopped at lower masses because of closed magnetic loops; here the loops are shown to be numerical artifacts of low resolution or of the filled mound profile. If correct, the result means low-field neutron stars can bury their fields within hours of accretion, supporting the accretion-reprocessing scenario for millisecond pulsars.

What carries the argument

The machinery is the axisymmetric, zero-toroidal-field Grad-Shafranov equation $\Delta_2 \psi = K(\psi,r,\theta)$ with $K = -4\pi r^2 \sin^2\theta\, \rho g\, dr_0/d\psi$, where $\psi$ is the poloidal flux function and $r_0(\psi)$ is a prescribed profile giving the height of each flux surface; for the ring-shaped mound, $r_0(\psi)$ is the inverted parabola of Eq. 16, which loads mass only up to the truncation angle $\theta_t$. The new ingredient is a multipolar current-free boundary condition (Eq. 15) that updates the outer radial boundary at every SOR iteration by decomposing the interior solution into associated Legendre multipoles ($\ell \le 33$) and matching to a source-free vacuum exterior, so the dipole and higher moments are free to evolve rather than being pinned to their surface values. This boundary condition is validated by showing that solutions become independent of the outer radius choice. The stretched radial coordinate ($y = \log((r-aR_*)/(R_*(1-a)))$ with $a=0.999$) and grid resolutions up to $5000^2$–$12000^2$ are what let the solver follow the large gradients at the mound edge and the subsequent equatorward spread.

What would settle it

Run the same GS solver while conserving $dM/d\psi$ between solutions, or perform a time-dependent axisymmetric MHD simulation of accretion onto a $10^9$ G neutron star with mass loaded up to $r_t = 0.6 R_A$, tracking the mound for hours: if the mound does not spread to the equator and the normalized dipole moment does not decline to about $0.627$, the burial result is an artifact of the assumed profile function.

Watch

Extended reading notes

Core claim

The paper's central claim is that magnetically confined accretion mounds, when computed at sufficient resolution with a boundary condition that lets multipoles evolve freely, do not hit a hard mass ceiling: instead the mound spreads equatorward as its mass grows, and this spreading is the physical mechanism of field burial. For a $10^9$ G neutron star with a mound loaded out to $r_t = 0.6 R_A$, the solution spreads all the way to the equator and the normalized dipole moment at the outer radius falls to $0.627$, a 37 percent reduction, with an octupole contribution rising to $0.12$. The paper argues this is the onset of field burial that earlier semi-analytic work anticipated, and that the closed magnetic loops which stopped previous solutions near $10^{-7}$–$10^{-8}$ M$\odot$ were numerical artifacts: they disappear at high resolution or with the ring-shaped profile, and the true limit is a resolution-dependent Numerical Maximum Mass beyond which solutions become non-unique. It also shows that the same physics applies when the mound sits on a pre-existing ocean, where sinking reduces ellipticity and dipole moment, and that quadru-dipolar surface fields make the two polar mounds asymmetric, with the asymmetry growing until matter funnels through the quadrupolar field.

Load-bearing premise

The equatorward spreading and the 37 percent dipole reduction depend on the hand-specified ring-shaped profile $r_0(\psi)$, which fixes the mass loading per flux surface; the paper explicitly does not impose a fixed $dM/d\psi$ between different solutions, so a physically self-consistent accretion model could yield a different profile and materially different burial efficiency.

Editorial extensions

If this is right

  • For a $10^9$ G neutron star with $\zeta=0.6$, a mound accreted in about 9 hours spreads to the equator and cuts the normalized dipole moment to $0.627$, implying short-term field burial on outburst timescales.
  • The resolution dependence of the Numerical Maximum Mass means published caps such as $10^{-7}$ M$\odot$ for a relativistic degenerate EOS are not physical limits; denser or better-resolved grids should push them higher.
  • Mass ellipticity grows with accreted mass, reaching $\sim 3.75\times10^{-10}$ at $10^{12}$ G, and turns over once a mound spreads to the equator, a concrete signature for continuous gravitational wave searches.
  • Mounds that sink into a pre-existing ocean retain most of their field-burial effect: with a helium composition and the crystallization-depth inner boundary, dipole reduction is slightly weaker and ellipticity is lower than for a pure hard-crust mound.
  • Quadru-dipolar surface fields create asymmetric polar mounds with different heights and masses at the two poles, and beyond a field-dependent quadrupole fraction ($f_{qd} \gtrsim 1$ at $10^{10}$ G, $\gtrsim 5$ at $10^{12}$ G) accretion funnels directly through the quadrupolar field, outside the model's validity.

Reading between the lines

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

  • The 37 percent burial figure is tied to the assumed $r_0(\psi)$; a self-consistent model that conserves $dM/d\psi$ between solutions could reduce or amplify the spreading, so the number should be read as a proof of mechanism rather than a precise prediction.
  • If the current-free boundary result transfers to full time-dependent MHD, the same equator-ward spreading might operate in accreting millisecond pulsars on timescales of hours, making magnetic field evolution observable in a single outburst rather than over Gyr.
  • The octupole enhancement (0.12 of the dipole) in the $10^9$ G burial case suggests that the surface field complexity inferred from X-ray pulse-profile modeling could be a direct signature of ongoing burial, not just an intrinsic multipole.
  • The quadru-dipolar funneling limit predicts a qualitative change in accretion geometry, hot spots near the equator rather than the pole, which could be searched for in X-ray light curves of accreting pulsars with known field geometries.
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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 / 4 minor

Summary. The paper solves the axisymmetric Grad-Shafranov (GS) equation for magnetically confined accretion mounds on neutron stars, using stretched spherical coordinates and a new multipolar current-free outer boundary condition. With a ring-shaped mound profile r0(psi) whose truncation angle is tied to the Alfven radius, the authors find that as mound mass increases, the mound spreads latitudinally toward the equator and the normalized dipole moment at the outer radius decreases. For B_d = 10^9 G and zeta = 0.6, the reported ratio is mu_1(Rout)/mu_1(R*) = 0.627, which they interpret as the onset of field burial. They also model mounds on a pre-existing ocean/envelope and quadrudipolar inner boundary configurations, which produce asymmetric polar mounds. The solver is validated against a Green's-function boundary condition (Appendix B3) and a perturbation-based uniqueness test defines a resolution-dependent Numerical Maximum Mass (NMM).

Significance. If the central result holds, the paper would overturn the common interpretation that closed magnetic loops set an intrinsic mass ceiling in GS mound calculations, and would support rapid field burial for low-field neutron stars in the early accretion phase. The new current-free boundary condition, the large parameter suite, the validation against a Green's-function volume boundary condition (relative difference below 5e-5 in Appendix B3), and the clearly described perturbation tests for numerical uniqueness are all valuable and reproducible contributions. However, because the headline dipole reduction is quoted at a resolution-limited numerical cutoff and the mass-loading profile is prescribed, the quantitative field-burial claim is not yet established as a converged physical prediction.

major comments (4)
  1. [Sec. 4.2.3, Table 5; Appendix C, Table C1] The central quantitative result, mu_1(Rout)/mu_1(R*) = 0.627 for B_d = 10^9 G and zeta = 0.6, is the value at the Numerical Maximum Mass, and NMM is not a physical mass limit. Table C1 shows that NMM increases monotonically with resolution, from 3.34e-13 M_sun at 700^2 to 5.65e-13 M_sun at 5000^2, and the text states that NMM is 'limited by resolution.' Because the dipole ratio decreases monotonically with accreted mass (Figs. 7 and 9), a higher-resolution run that supports a larger NMM could shift the ratio substantially below 0.627. The paper does not report how mu_1(Rout)/mu_1(R*) itself converges with resolution for this case, so the specific '37% reduction' is not a converged numerical prediction.
  2. [Sec. 2.5, Eq. (16); Limitations (ii)] The profile function r0(psi) in Eq. (16) is prescribed, and the limitations acknowledge that no fixed mass-loading per flux tube dM/dpsi is imposed between different solutions. Consequently, the sequence of solutions in Fig. 6 and the mass-dipole relations in Figs. 8-10 are not a single accretion sequence: each snapshot is an independent equilibrium with a different dM/dpsi. The equator-ward spreading and the 0.627 dipole ratio therefore measure properties of the chosen profile family, not of a physically self-consistent loading history. A model that fixes dM/dpsi during mound growth, or a detailed justification that the adopted r0(psi) follows from disk-magnetosphere interaction, is needed to support the field-burial interpretation.
  3. [Sec. 4.2.1; Limitations (v)] The text notes that matter beyond theta_t is supported by vacuum magnetic fields (or by a low-density envelope in the ocean model) and states that the dynamical stability of these spreading mounds should be verified. Since the central claim is that a mound can persist while spreading and burying the field, a static GS equilibrium alone does not demonstrate field burial unless the configuration is stable or the instability growth time exceeds the accretion time. Given the known interchange and pressure-driven instabilities of such mounds, the burial result should be presented conditional on a stability check.
  4. [Limitation (iv)] The authors report that general-relativistic GS solutions show approximately three times less screening than the Newtonian solutions (Rossetto et al. 2023). This directly affects the quantitative claim: the 37% reduction is a Newtonian value and may overestimate burial for the same mound parameters. The limitation is acknowledged, but the abstract and Section 4 present the Newtonian number without this caveat; the discussion in Section 6 should carry the GR caveat whenever the 0.627 figure is quoted.
minor comments (4)
  1. [Table 3] The column header 'R_b (km)' lists values such as 10, 9.98, and 9.93, which could be confused with R*; clarify that the dipole moment ratio in the last column is measured with respect to the inner boundary R_b, not the stellar surface.
  2. [Eq. (29)] The symbols alpha, epsilon, and h/r are used before their adopted values are introduced; the text should define these disk parameters explicitly when they first appear.
  3. [Eqs. (23), (25), (26)] These fits are empirical descriptions of the authors' own computed points; the text should state explicitly that they are not derived relations and should not be extrapolated outside the fitted mass and height ranges.
  4. [Sec. 4.1] The phrase 'Masses with an order of 10^-8 M_sun can be simulated' should read 'masses of order 10^-8 M_sun' to avoid grammatical ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dipole-moment ratios and ellipticities are computed outputs of the solved Grad–Shafranov equilibrium, not restatements of the prescribed profile functions.

full rationale

The derivation chain is: choose the EOS, the inner boundary field, the outer current-free boundary condition, the profile function r0(psi) (e.g. Eq. 16), and the mound height rc; solve Eq. 11 iteratively; then compute psi, from which Eq. 20 gives multipole moments and Eq. 22 gives ellipticity. The field-burial claims are readings of these converged numerical outputs (Figs. 6, 9, Table 5), not identities with the inputs. The hand-specified r0(psi) is an explicit modelling assumption, and the paper itself states the limitation that no fixed dM/dpsi between solutions is imposed, so the output mass distribution is not a hidden re-description of the input. The empirical fits in Eqs. 23, 25, and 26 are descriptive scalings of already-computed data and do not feed back into the central result. Self-citations to Mukherjee (2017) supply the ring-shaped profile and the Paczynski EOS as provenance for model choices; they are not invoked as an external uniqueness theorem that forces the outcome. The new current-free boundary condition is validated internally against a Green's-function volume boundary condition (Appendix B3) and by demonstrated domain-size independence (Figs. 3–4), so the boundary treatment is not circularly importing the dipole suppression. The resolution dependence of the Numerical Maximum Mass (Appendix C, Table C1) is an acknowledged numerical limitation that affects robustness of the quoted 37% reduction, but it is not a circularity: the NMM is defined by a perturbation-based uniqueness test applied to the solver, not by the quantity being predicted. No load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 11 free parameters · 8 assumptions · 0 invented entities

The central claim depends on a small set of physical assumptions: axisymmetric poloidal magnetostatics, Newtonian gravity, Paczynski EOS, and a prescribed mound profile. The profile parameters and empirical fit coefficients are the main free inputs. No new physical entities are introduced.

free parameters (11)
  • Ring profile shape constants (0.25, 0.5) = 0.25, 0.5
    The ring-shaped mound profile r0(psi) = R* + rc/0.25 * [0.25 - (psi/psi_a - 0.5)^2] (Eq. 16) uses hand-chosen coefficients to create a hollow mound; these constants are ad hoc inputs not derived from data.
  • Maximum mound height r_c = Variable per run (e.g., 4.7 m to 103.1 m)
    Maximum mound height is a scan parameter; central field burial results are computed as a function of r_c up to the numerical maximum mass.
  • Truncation angle theta_t = Computed from Eq. 17 via Alfvén radius, or arbitrary (e.g., 50 deg)
    Angular extent of mass loading is either derived from the disk truncation radius r_t = zeta R_A, or arbitrarily chosen to probe large masses; it directly affects dipole moment reduction.
  • Truncation radius fraction zeta = 0.6, 0.7, 0.8, 0.9, 1.0
    Ratio r_t/R_A, motivated by MHD simulations; varies the angular extent and accreted mass for each B_d.
  • Envelope depth h_env = 0 to 70 m
    Free parameter in the ocean profile (Eq. 24) used to approximate sinking; the results in Table 3 and Eqs. 25-26 depend on its values.
  • Poleward boundary parameter psi_ap = Corresponds to 4 deg
    Arbitrary poleward boundary of the mound-on-ocean profile, chosen from light cylinder constraints as a conservative estimate.
  • Quadrupole to dipole fraction f_qd = 0.1, 1.0, 5.0
    Surface quadrupole to dipole ratio; varied to explore asymmetric mounds and determines the truncation angles via Eqs. 29-30.
  • Fit coefficients for dipole moment vs mass (Eq. 23) = 0.994, 0.00386, 1.376
    Empirical power-law fit to the authors' numerical data for arbitrary theta_t; descriptive of the trend, not predictive.
  • Fit coefficients for dipole moment and ellipticity vs h_env (Eqs. 25, 26) = 0.9739, 2.02e-8, 3.058 and 2.8835, 1.5e-5, 2.145
    Empirical fits to the sinking sequence; illustrative of trends, not independently measured.
  • Grid stretching parameter a = 0.999
    Chosen to resolve the mound (Appendix A); affects numerical resolution but not the physical input.
  • Quadrudipolar disk parameters alpha, h/r, epsilon = 0.1, 0.1, 1.0
    Assumed from Çıkıntoğlu (2023) for the Alfvén radius estimate in the quadrudipolar cases (Eq. 29).
assumptions (8)
  • domain assumption Axisymmetry and zero toroidal magnetic field
    The field is written as B = grad(psi) x phi_hat / (r sin theta) with B_phi = 0 (Sec. 2.1), restricting to 2D poloidal equilibria.
  • domain assumption Constant gravitational acceleration g = G M* / R*^2
    Assumed in Eq. 3; valid for thin mounds but not for heights comparable to R*.
  • domain assumption Newtonian gravity
    GR effects are not included; the paper cites Rossetto et al. (2023) showing GR reduces screening by about a factor of 3.
  • domain assumption Paczynski EOS for zero-temperature degenerate electron gas
    Approximates pressure over the mound density range (Eq. 12); assumes pure ionized helium (mu_e = 2.0).
  • domain assumption Fixed dipolar or quadrudipolar inner boundary
    The crust is assumed to hold the field fixed; in reality the field may sink into the ocean, which the paper acknowledges.
  • ad hoc to paper Prescribed analytical profile function r0(psi)
    The mound shape and mass distribution are inputs (Eqs. 16, 21, 24), not derived from accretion dynamics; dM/dpsi is not conserved between solutions (acknowledged limitation).
  • ad hoc to paper Current-free vacuum above R_in for the outer boundary condition
    The CFB (Eq. 15) assumes a source-free region above R_in and multipole convergence with lmax=33; validated by comparison with a volume Green's function (Appendix B3).
  • domain assumption Quadrudipolar Alfvén radius prescription from Çıkıntoğlu (2023)
    Used to compute theta_t for quadrudipolar cases (Eqs. 29-30); disk parameters alpha, h/r, epsilon are assumed.

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Pith. "Pith review of Investigating field burial by magnetically confined accretion mounds on Neutron Stars." pith.science (2026). https://pith.science/paper/ZIC2E7IG

@misc{pith2026250708509,
  author       = {Pith},
  title        = {Pith review of: Investigating field burial by magnetically confined accretion mounds on Neutron Stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIC2E7IG}},
  note         = {Machine review of arXiv:2507.08509}
}
read the original abstract

We explore the problem of magnetic confinement of accreted matter forming an accretion mound near the magnetic poles of a neutron star. We calculate the magnetic field geometry of the accreted mound by solving the magnetostatic Grad Shafranov (GS) equation in radially stretched spherical coordinates with high resolution and an extended domain. In this work, we propose a new physically motivated multipolar current free boundary condition at the outer radial boundary. We have evaluated a large suite of GS solutions for different neutron star magnetic fields and mound configurations. We find that with sufficient resolution, the ring-shaped mound profiles spread latitudinally on the neutron star surface, towards the equator, with a potential decline in dipole moment at outer radii, demonstrating the onset of field burial. A higher latitudinal spread towards the equator leads to more effective magnetic field burial. Along with the ring-shaped mound profile on a hard crust majorly used in this work, we also model mounds formed on a pre-existing ocean, which is more physically motivated. Additionally, we explore different GS solutions for a quadru-dipolar surface magnetic field. We find that such configurations lead to asymmetric polar mounds. We discuss the validity of such solutions for different relative strengths of the quadrupole and dipolar components.

Figures

Figures reproduced from arXiv: 2507.08509 by the authors.

Figure 2
Figure 2. Plots of magnetic field lines for fixed dipole condition (red dashdot lines) and CFB (black solid lines) at the outer radial boundary for Case 1 : 𝐵𝑑 = 109 G, 𝜃𝑡 = 45.5 0 , 𝑟𝑐 = 5.5 m, 𝑀 = 5.42 × 10−13 M⊙ (top) and Case 2 : 𝐵𝑑 = 1012 G, 𝜃𝑡 = 84.0 0 , 𝑟𝑐 = 103.1 m, 𝑀 = 1.47 × 10−8 M⊙ (bottom). Magnetosphere gets compressed as dipole moment is lowered due to the accretion mound. Solutions for a larger mass (due to a h… view at source ↗
Figure 1
Figure 1. Plot of Density Profile and Magnetic field lines (solid) for param￾eters 𝐵𝑑 = 109 G, 𝜃𝑡 = 45.5 0 , 𝑟𝑐 = 4.7 m, 𝑀 = 2.952 × 10−13 M⊙ (top), colormap of magnetic field magnitude B (in Gauss) (middle) and colormap of𝑌dip (Eqn 19) (bottom). Dashed lines in the top figure are dipolar magnetic field lines. The accretion mound changes the magnetic field at the surface of the neutron star to a range of 108 − 1011 Gauss. Bot… view at source ↗
Figure 3
Figure 3. Plots of magnetic field lines for a small radial domain 2 km (solid lines) and a large radial domain 12 km (dashdot lines) with a Fixed Dipole Boundary Condition (top), Outflow Boundary Condition (middle) and CFB (bottom) at the outer radial boundary for Case 2 in subsection 3.2.1. The inset plots show the same plots in logarithmic scale. Magnetic field line solutions for CFB seems to be independent of the choice of… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Normalized Dipole Moment at a radius above the neutron star relative to the normalized dipole moment at the surface of the neutron star (Equation 20) plotted for solutions with three boundary conditions and two different simulation domains i.e for six cases. Vertical l…
Figure 5
Figure 5. Figure 5: Density profiles and magnetic field lines (solid) for the 1/cosh profile for 𝑟𝑐 = 216 m (top) and 𝑟𝑐 = 227 m (bottom). These solutions are calculated for a 𝐵𝑑 = 1012.5 G. Dashed lines are the undistorted dipolar magnetic field lines. Masses with an order of 10−8 M⊙ can…
Figure 6
Figure 6. Figure 6: Density profiles and magnetic field lines (solid) for the ring-shaped mound profile for parameters 𝐵𝑑 = 1010 G, 𝑟𝑡 = 0.7𝑅𝐴, 𝜃𝑡 = 26.210 (top plot) and parameters 𝐵𝑑 = 1012 G, arbitrary 𝜃𝑡 = 500 (bottom plot). The top plot X-axis has a 𝜃 range of 0 0 − 500 , while the b…
Figure 7
Figure 7. Figure 7: Top plot has absolute mass ellipticity of the 2 mounds on the Y-axis and mass of the two mounds on the X-axis for four values of 𝐵𝑑. Ellipticity is proportional to the accreted mass. Bottom plot has normalized dipole moment at the outer radius relative to its value at …
Figure 8
Figure 8. Figure 8: Plot of Mass (in 10−12 M⊙) for 5 different possible truncation angles (𝜃𝑡) made for 4 surface magnetic field strengths (𝐵𝑑). The colored symbols indicate the maximum mass (NMM) set by this simulation for a respective 𝜃𝑡 , while the gray symbols are the solutions for a …
Figure 9
Figure 9. Figure 9: Plot of normalized dipole moment at the outer radius relative to its value at the NS surface for 5 different truncation angles (𝜃𝑡) made for 4 surface magnetic field strengths (𝐵𝑑). The colored symbols indicate the Maximum Mass (NMM) set by this simulation for a respec…
Figure 11
Figure 11. Figure 11: Density profile of the mound from Equation 24 for parameters 𝐵𝑑 = 1012 G, arbitrary 𝜃𝑡 = 500 , arbitrary 𝜓ap = sin2 (4.0 0 ), 𝑟𝑐 = 90 m, ℎenv = 30 m, accreted mass = 3.15 × 10−9 M⊙. Magnetic field lines for 𝜓ap,𝜓bp,𝜓bt and 𝜓𝑎 are also plotted here. The five regions se…
Figure 10
Figure 10. Figure 10: The parameters for all the three plots are 𝐵𝑑 = 1012 G and an arbitrary 𝜃𝑡 = 500 . The topmost plot has shown absolute ellipticity as a function of Mass. The middle plot has shown normalized dipole moment at the outer radius relative to its value at the neutron star s…
Figure 12
Figure 12. Figure 12: Absolute Ellipticity versus Envelope height (top plot) and Nor￾malized dipole moment at the outer radius with respect to the surface versus Envelope height (bottom plot). Absolute ellipticity reduces and relative dipole moment decreases with increase in the envelope d…
Figure 14
Figure 14. Figure 14: Polar Plot of magnetic field lines for three different surface quadrupolar field to surface dipolar field ratios. Black lines are the magnetic field lines with positive 𝜓, while the red magnetic field lines are opposite to the black lines with negative 𝜓. Blue dotted …
Figure 15
Figure 15. Figure 15: Plot of Density Profile and Magnetic field lines (solid) for parameters 𝐵𝑑 = 1010 G and an initial quadrudipolar magnetic field of the neutron star. Plots have been made for Case B (left) and Case D (right). The quadrupole to dipole fractions ( 𝑓qd) have been indicate…
Figure 16
Figure 16. Figure 16: Plot of Density Profile and Magnetic field lines (solid) for parameters 𝐵𝑑 = 1012 G and an initial quadrudipolar magnetic field of the neutron star. Plots have been made for Case E (top middle) and Case F (bottom middle). Red lines are magnetic field lines with negati…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.