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ROTATING NEUTRON STAR MODELS WITH MAGNETIC FIELD

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M. Bocquet, S. Bonazzola, E. Gourgoulhon, J. Novak

General Relativity and Quantum Cosmology · Submitted 25 Mar 1995 · 1 reference

Pith's verdict: The first self-consistent relativistic magnetized rotating neutron star models, and the methods hold up; the headline 13-29% mass increase is a number for one current profile, not a generic result.

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arXiv IDgr-qc/9503044 (version 1)
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CitationE. Gourgoulhon, J. Novak, M. Bocquet, S. Bonazzola. "ROTATING NEUTRON STAR MODELS WITH MAGNETIC FIELD." arXiv:gr-qc/9503044 (1995). https://pith.science/paper/23DADKJT
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The paper

The paper claims

Coupled Einstein-Maxwell numerical models show that poloidal magnetic fields up to about 10^14 T raise the maximum neutron star mass by 13 to 29 percent and increase the maximum rotation rate.

The reading

Neutron stars are among the densest objects with a surface, and many carry magnetic fields a trillion times stronger than Earth's. Until this paper, computer models of how such a star holds itself up against gravity treated it as a pure fluid: the magnetic field was either ignored or added afterwards as a small correction. This paper does the harder thing. It solves the equations that couple gravity (Einstein's equations), electromagnetism (Maxwell's equations) and the conducting fluid all at once, for a star that is spinning rapidly. The method is spectral: the star's interior and the surrounding vacuum are represented by smooth basis functions rather than a grid, and the whole system is iterated until it stops changing. The authors check the electromagnetic half of the code against three cases with known exact answers: a rotating magnetic dipole in flat space, Ferraro's 1954 analytic solution for a magnetized Newtonian fluid, and two virial identities that any true solution of Einstein's equations must satisfy. The physics result is that the magnetic field acts like a centrifugal force. It flattens the star, and at extreme field strengths it pinches the poles inward. The largest field the code can sustain has a polar value near 10^14 T, roughly where the magnetic pressure at the centre matches the fluid pressure. At that limit the maximum mass of a non-rotating star rises by 13 to 29 percent depending on which equation of state is used, and the maximum spin rate also rises. Two caveats matter. The field geometry is not predicted; it is chosen by hand through a current function, and the mass-increase numbers come from one particular choice. And the upper field limit is where the iteration stops converging, which the authors read as the absence of any equilibrium.
Extended reading notes

Core claim

The paper asserts that the maximum gravitational mass and the maximum rotation rate of neutron stars both increase with the poloidal magnetic field, and quantifies it: at the largest field the code supports, the static maximum mass rises by 28.6 percent for Pol2, 20.2 percent for DiazII, 14.9 percent for PandN, 17.3 percent for BJI and 13.3 percent for HKP relative to the non-magnetized maximum mass (Table 1, third line versus first line; abstract s10). If correct, this establishes that magnetic fields, not only rotation, can raise the neutron star mass limit, and that the effect is large enough to matter for the interpretation of heavy compact objects.

Load-bearing premise

The entire mass-increase result is computed along sequences at fixed current function f(A_phi) = const (Eq. 33, Sect. 4.3 and Table 1). The current function is a free functional input, not derived from any microphysics, and it fixes the field geometry and the spatial distribution of the Lorentz force. The paper does test two other forms, Eqs. (35) and (36), but only to display the resulting field structure (Figs. 14 to 16); the maximum-mass tables and the 13 to 29 percent figures are never recomputed for those alternatives. So the headline number is a property of one hand-picked one-parameter family, and the paper does not show it is generic across admissible current distributions.

Original abstract

We present the first numerical solutions of the coupled Einstein-Maxwell equations describing rapidly rotating neutron stars endowed with a magnetic field. These solutions are fully relativistic and self-consistent, all the effects of the electromagnetic field on the star's equilibrium (Lorentz force, spacetime curvature generated by the electromagnetic stress-energy) being taken into account. The magnetic field is axisymmetric and poloidal. Five dense matter equations of state are employed. The partial differential equation system is integrated by means of a pseudo-spectral method. Various tests passed by the numerical code are presented. The effects of the magnetic field on neutron stars structure are then investigated, especially by comparing magnetized and non-magnetized configurations with the same baryon number. The deformation of the star induced by the magnetic field is important only for huge values of B (B>10^{10} T). The maximum mass as well as the maximum rotational velocity are found to increase with the magnetic field. The maximum allowable poloidal magnetic field is of the order of 10^{14} T (10^{18} G) and is reached when the magnetic pressure is comparable to the fluid pressure at the centre of the star. For such values, the maximum mass of neutron stars is found to increase by 13 to 29 % (depending upon the EOS) with respect to the maximum mass of non-magnetized stars.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,125 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.46 · human participants 0.01 · controlled comparison 0.89 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.79 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

We present the first numerical solutions of the coupled Einstein-Maxwell equations describing rapidly rotating neutron stars endowed with a magnetic field. These solutions are fully relativistic and self-consistent, all the effects of the electromagnetic field on the star's equilibrium (Lorentz force, spacetime curvature generated by the electromagnetic stress-energy) being taken into account. The magnetic field is axisymmetric and poloidal. Five dense matter equations of state are employed. The partial differential equation system is integrated by means of a pseudo-spectral method. Various tests passed by the numerical code are presented. The effects of the magnetic field on neutron stars structure are then investigated, especially by comparing magnetized and non-magnetized configurations with the same baryon number. The deformation of the star induced by the magnetic field is important only for huge values of B (B>10^{10} T). The maximum mass as well as the maximum rotational velocity are found to increase with the magnetic field. The maximum allowable poloidal magnetic field is of the order of 10^{14} T (10^{18} G) and is reached when the magnetic pressure is comparable to the fluid pressure at the centre of the star. For such values, the maximum mass of neutron stars is found to increase by 13 to 29 % (depending upon the EOS) with respect to the maximum mass of non-magnetized stars.

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 numerical solutions of the coupled Einstein-Maxwell equations describing rapidly rotating neutron stars endowed with a magnetic field. contribution 1.00 0.97 0.12 asserted 1.00
s1 These solutions are fully relativistic and self-consistent, all the effects of the electromagnetic field on the star's equilibrium (Lorentz force, spacetime curvature generated by the electromagnetic stress-energy) being taken into account. contribution 0.87 0.87 0.06 asserted 1.00
s2 The magnetic field is axisymmetric and poloidal. contribution 0.48 0.60 0.02 asserted 1.00
s3 Five dense matter equations of state are employed. method 0.99 0.77 0.02 asserted 0.99
s4 The partial differential equation system is integrated by means of a pseudo-spectral method. method 1.00 0.90 0.02 asserted 1.00
s5 Various tests passed by the numerical code are presented. method 0.90 0.88 0.02 asserted 0.97
s6 The effects of the magnetic field on neutron stars structure are then investigated, especially by comparing magnetized and non-magnetized configurations with the same baryon number. contribution 0.80 0.91 0.08 asserted 0.86
s7 The deformation of the star induced by the magnetic field is important only for huge values of B (B>10^{10} T). result 0.99 0.89 0.11 asserted 1.00
s8 The maximum mass as well as the maximum rotational velocity are found to increase with the magnetic field. result 1.00 0.96 0.26 asserted 1.00
s9 The maximum allowable poloidal magnetic field is of the order of 10^{14} T (10^{18} G) and is reached when the magnetic pressure is comparable to the fluid pressure at the centre of the star. result 1.00 0.90 0.20 asserted 1.00
s10 For such values, the maximum mass of neutron stars is found to increase by 13 to 29 % (depending upon the EOS) with respect to the maximum mass of non-magnetized stars. result 1.00 0.96 0.30 asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

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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. Comparison between the exact solution and the numer￾ical one in the case of a rotating conducting sphere with a point magnetic dipole moment at its centre. The left figure corre￾sponds to the electric field, the error being defined as the square root of [(E(r) − E exact (r) ) 2 + (E(θ) − E exact (θ) ) 2 ]/[(E(r)) 2 + (E(θ)) 2 ]. The right figure depicts the same thing for the magnetic field. Asterisks denote the sphere’s surface. 3.2.2. Comparison with Ferraro’s solution Ferraro (1954) has obtained an analytical solution for a Newtonian incompressible fluid body with an ele
Figure 2. Comparison between Ferraro’s analytical solution and the numerical one in the case of a Newtonian incompressible fluid endowed with a magnetic field corresponding to the cur￾rent function f(x) = const. The plotted quantity is the relative difference between the two values of Aφ as a function of the radial coordinate r for three values of θ. Asterisks denote the star’s surface. value of the GRV2 or GRV3 error indicator is of the order 10−5 (polytropic EOS) or a few 10−4 (tabulated EOS). Let us recall that tabulated EOS introduce an important numerical error because they do not stric
Figure 3. Fluid proper density isocontours in the (r, θ) plane of a static magnetized star of 2.98 M⊙ built on the Pol2 EOS of Salgado et al. (1994a) (polytropic EOS with γ = 2) with a current function f(x) = const. = 4 × 1015/R A m−2 ρ −1 nucc −2 , where R is the r coordinate of the star’s equator. The central energy density is e = 1.42 ρnucc 2 (1 ρnuc := 1.66×1017 kg m−3 ). 4.1. Deformation of the star Even in the static case, the stress-energy tensor of the elec￾tromagnetic field is not isotropic (cf. Eq. (5.33) of BGSM with U = 0) so that the star deviates from spherical sym￾metry. For…
Figure 5. Fluid proper density isocontours in the (r, θ) plane for the maximum mass static magnetized star built on the Pol2 EOS of Salgado et al. (1994a) (polytropic EOS with γ = 2). The current function f(x) = const. = 5.63 × 1015/R A m−2 ρ −1 nucc −2 , where R is the r coordinate of the star’s equator. The central energy density is e = 1.42 ρnucc 2 and the mass M = 4.06 M⊙; the other char￾acteristics are listed on the third line of
Figure 6. Magnetic field lines in the (r, θ) plane for the maximum mass configuration corresponding to
Figure 8. Maximum gravitational mass (left) and maximum baryon number (right) along static sequences at fixed magnetic dipole moment as a function of the magnetic dipole moment for the DiazII EOS . 10
Figure 10. Electric current distribution induced by the current function f(x) = const. = 1015/R A m−2 ρ −1 nucc −2 for a Pol2 EOS model, of central energy density ec = 3.06 ρnuc c 2 , rotating at Ω = 3 × 103 rad s−1 . The plotted quantity is the value j (φ) of the azimuthal component of j in an orthonormal basis in the equatorial plane θ = π/2 (solide line), along θ = π/4 (dashed line) and along the rotation axis θ = 0 (dotted line). Asterisks denote the star’s equatorial radius. centre, the surface value being approximatively four times lower. The dominant dipolar structure of B clearly…
Figure 11. Components B(r) (left) and B(θ) (right) of the mag￾netic field generated by the electric current distribution of
Figure 12. Components E(r) (left) and E(θ) (right) of the electric field corresponding to the configuration of Figs. 10 and 11. The notations are the same as in
Figure 13. Magnetic field lines (left) and electric isopotential lines At = const (right) in the (r, θ) plane for the configuration considered in Figs. 10, 11 and 12. The thick line denotes the star’s surface. where A0 φ = 4π × 104 R3 T m−1 , R being the value of the coordinate r at the equator. One might think about other choices, such as f(x) = αx + β, f(x) = αx2 , or f(x) = α/(1 + x 2 ), but with such functions, the iterative procedure described in Sect. 3.1 revealed not to converge, resulting in a solution either cyclic or chaotic with re￾spect to the number of steps. This behaviour is…
Figure 16. Components E(r) (left) and E(θ) (right) of the electric field corresponding to the configuration of Figs. 14 and 15. The notations are the same as in
Figure 21. Maximum gravitational mass (left) and maximum baryon number (right) along rotating sequences at fixed magnetic dipole moment as a function of the magnetic dipole moment for the PandN EOS. 15
Figure 25. Differences between configurations without magnetic field and with a magnetic dipole moment M = 1.5 1032 A m2 along a sequence at constant baryonic mass B = 3.00 M⊙ for the Pol2 EOS and the current function f(x) = const: (a) gravitational mass difference ∆M; (b) angular momentum difference ∆J; (c) circumferential radius difference ∆Rcirc; (d) central energy density difference ∆ec. The path along the sequence is parametrized by the angular velocity Ω. origin, not in a high rotation rate, but in a large spa￾tial extension. The Lorentz forces play then a greater role in the…

Showing 13 of the paper's 26 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9503044 by the authors.

References
How the literature uses it

34 arXiv papers cite this in 73 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 50 sentences without a confident label

Uses its method or tool 7 papers
  • In this work, we construct equilibrium configuration of magnetized stellar models by using the same approach as in [ref] .

    Crust effects and the cooling relaxation time in highly magnetized neutron stars · Introduction · 1.00

  • According to Ref. [ref] , the inhomogeneous Maxwell equation: ∇_μ F^αμ = μ_0j^α, can be expressed in terms of the two non-vanishing components of the electromagnetic potential $A_μ$ through the Maxwell-Gauss equation: [equation] and through the Maxwell-Ampère equation as: [equation] with the notation: [equation] Within the $3+1$ decomposition and under the assumptions of stationary and axisymmetric space-time, the Einstein equations for the metric potentials in Eq.5 are given by [ref] : [equation] being $E$ total energy density of the fluid given by: E= Γ^2 (e + p) - p,

    The internal composition of proto-neutron stars under strong magnetic fields · General Relativistic Calculation · 0.87

  • In this section we present the framework and assumptions made to compute numerical models of magnetized neutron stars, including the possibility of strong deformation, using the code [ref] and the lorene library [ref] .

    Magnetized neutron stars: perturbative versus fully-numerical approaches · Full numerical solutions · 1.00

Compares with it 1 paper
  • Some interesting aspects arise from these solutions: firstly the mass of the star decreases monotonically with $q$; this is the opposite behavior to that obtained in models of neutron stars with magnetic fields [ref] , where their structure begins from spherical morphology at zero magnetic field (for the static case), and flattens, increasing the circumferential radius of the star, as the magnitude of the magnetic fields increase, with a corresponding increase in the mass of the star.

    Magnetostatic boson stars · Sequence of magnetic boson stars · 1.00

Cites it as background 10 papers

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