Structural quasi-universality in highly magnetized differentially rotating neutron stars

Pratul Manna; Surajit Kalita

REVIEW 4 major objections 5 minor 56 references

Structural quasi-universality in highly magnetized differentially rotating neutron stars

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A quasi-universal relation connects moment of inertia and compactness in neutron stars even when they rotate differentially and carry magnetic fields up to 10^18 G.

desk verdict A useful extension of Breu-Rezzolla to magnetized, differentially rotating stars, but the printed central equation is dimensionally inconsistent and the A-dependence is overfit. read the letter →

arxiv 2608.02824 v1 pith:J27JNVWV submitted 2026-08-03 astro-ph.HE astro-ph.SRgr-qc

Pratul Manna , Surajit Kalita This is my paper · ORCID

classification astro-ph.HEastro-ph.SRgr-qc

keywords quasi-universalrelationsneutronstarsmomentofinertiacompactnessdifferentialrotationtoroidalmagneticfieldsmagnetarsequationstate

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 argues that the normalized moment of inertia $\tilde I = I/M^3$ of a neutron star is set almost entirely by its compactness $C = M/R$, even for configurations that rotate differentially and carry strong toroidal magnetic fields. This extends a previously known slow-rotation universal relation by allowing the fit coefficients to depend on the differential-rotation parameter $A$ and on magnetic field strength through the dimensionless ellipticity $\epsilon = B_{\max}^2 R^4 / M^2$. Using 600 equilibrium models built with ten hadronic equations of state, the paper fits a single phenomenological formula with average relative error of about 8% and maximum relative error of about 50%. If the relation holds, it yields equation-of-state-insensitive estimates of neutron-star structure and shows that ignoring differential rotation and magnetic fields biases magnetar spin-down luminosities and post-merger rotational energies.

What carries the argument

The load-bearing object is the additive decomposition $\tilde I(C,A,B_{\max}) = \tilde I_0(C,A) + \Delta \tilde I(C,B_{\max})$, where the rotational part keeps the inverse-power form in $C$ and the magnetic part is written as a polynomial in $C/\sqrt{\epsilon}$ with $\epsilon = B_{\max}^2 R^4/M^2$. This scaling variable collapses the magnetic contribution onto a monotonic curve and forces the correction to vanish as $B_{\max} \to 0$, so the non-magnetized universal relation is recovered exactly. The coefficients of both polynomials carry all the dependence on differential rotation, parametrized by the length scale $A$ of the $J$-constant rotation law.

What would settle it

Build equilibrium models with poloidal or mixed magnetic field configurations at $B_{\max}$ up to $10^{18}$ G using a solver that permits them, and compare their $\tilde I$–$C$ curves to Eq. (12); deviations larger than the paper's roughly 50% maximum relative-error band, or a failure of the additive decomposition, would refute the claimed quasi-universality.

Watch

Extended reading notes

Core claim

The central claim is that the relation $$\tilde I(C,A,B_{\max}) = \sum_{i=1}^4 \left[a_i(A) + b_i(A)\left(\frac{B_{\max}$R^{2}$}{\sqrt{G M}}\right)^i\right] $C^{{-i}}$$$ remains valid for uniformly rotating, differentially rotating, and highly magnetized neutron stars alike, provided the sequences have constant angular momentum and lie on the stable branch with $j = J/M^2 \lesssim 0.35$. The coefficients $a_i(A)$ and $b_i(A)$ are fitted third-order polynomials in inverse powers of $A$, recovering the uniform-rotation limit as $A \to \infty$. The paper demonstrates the relation by reproducing its 600 numerical configurations to within an average relative error of about 8% and a maximum of about 50%, with no strong dependence on the underlying equation of state. It then applies the relation to show that magnetic-field corrections can change inferred magnetar spin-down luminosities by orders of magnitude and that the rotational kinetic energy of the GW170817 remnant is constrained near $10^{53}$ erg, with frequency alone unable to break the degeneracy between differential rotation and magnetic field strength.

Load-bearing premise

The magnetic correction is assumed, not derived, to separate additively from rotation and to follow the chosen polynomial in $C/\sqrt{\epsilon}$, and it is calibrated only for purely toroidal fields.

Editorial extensions

If this is right

The same fitting formula can be used to estimate $\tilde I$ for a neutron star when only $M$, $R$, $B_{\max}$, and the rotation law are known, regardless of which of the ten equations of state is correct.
Magnetar spin-down luminosities inferred from $P$ and $\dot P$ shift by orders of magnitude once the magnetic-field-corrected moment of inertia replaces the canonical value, so catalog comparisons need a structural correction.
For a fixed $C$, stronger differential rotation lowers $\tilde I$, meaning constant-angular-momentum sequences with the same compactness have systematically smaller moments of inertia than uniformly rotating stars.
Post-merger remnant analyses that assume uniform rotation and weak magnetization will misestimate rotational kinetic energy; constraining the spin frequency alone leaves the differential-rotation parameter and the magnetic field strength degenerate.

Reading between the lines

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

If the same quasi-universality holds for poloidal or mixed magnetic geometries, the relation would let observers infer interior field strengths of magnetars from moment-of-inertia measurements without knowing the equation of state.
The success of the $C/\sqrt{\epsilon}$ scaling hints that a virial-type argument, not just a polynomial fit, may underlie the magnetic correction; deriving it would turn the phenomenological formula into a predictive relation.
A natural stress test is to evolve the same constant-$J$ sequences with dynamical magnetic braking: because the relation assumes conserved angular momentum, systems losing $J$ through winds or gravitational waves should deviate in a way the current formula does not capture.
Combining the relation with a second equation-of-state-insensitive observable, such as tidal deformability, may break the degeneracy between differential rotation and magnetic field strength that spin frequency alone cannot resolve.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper extends the Breu-Rezzolla I-C universal relation to toroidally magnetized, differentially rotating neutron stars on constant-J sequences. That is a new and plausible extension. But the headline Eq. (12) has a dimensional inconsistency that, as printed, does not follow from the paper's own Eq. (9). The stress-test note is correct. With that fixed, the paper is a decent numerical fitting study worth a serious referee.

What is good: it uses 10 EoSs and the XNS code to map a parameter space (A = 6, 8, 10, infinity; Bmax 10^17 to 10^18 G) that has not been systematically covered before. The writing is clear, and the limitations section is unusually honest about toroidal-only fields, zero temperature, and the phenomenological nature of the fits. The applications to magnetar spin-down and GW170817 are appropriately framed as proof-of-concept.

Where it gets soft. First, Eq. (12) as printed: substituting epsilon = Bmax^2 R^4/M^2 into Eq. (9) gives a magnetic coefficient (Bmax R^2/M)^i, not (Bmax R^2/sqrt(GM))^i. With G=c=1, sqrt(GM)=sqrt(M) in geometric units, so the bracket is dimensionful and the equation overstates the magnetic term by M^(i/2). For M=1.4 solar masses that is factors ~1.18, 1.40, 1.65, 1.96 for i=1..4. The validation in Fig. 4 presumably used the correct form, so the printed equation does not match the actual fitted relation. It has to be corrected.

Second, the A-dependence is built from four-parameter cubics through only four A values. That is interpolation, not validation, and the reported reduced chi-squared values near zero are meaningless for testing the functional form. A referee should ask for either more A values or a simpler form with fewer parameters.

Third, the magnetic correction is a residual fit: subtract the rotational fit, then fit the leftover. The maximum relative error of about 50% is large, and the relation is tested only for purely toroidal fields, which the authors themselves note are unstable. That limits claims of universality.

Fourth, no error bars are propagated anywhere, and the magnetar application uses a tuning parameter eta to map B_s to B_max, which can swing the spin-down luminosity by orders of magnitude.

Bottom line: the core idea is sound and the numerical work appears careful, but the printed central equation is wrong and the A-dependence is overfit. This deserves peer review, but it needs fixing before publication.

Referee Report

4 major / 5 minor

Summary. The paper uses the XNS code to construct stationary, axisymmetric equilibrium configurations of neutron stars with 10 hadronic equations of state, differential rotation described by the J-constant law, and purely toroidal magnetic fields, all on fixed-angular-momentum sequences. It proposes a unified phenomenological relation, Eq. (12), between the normalized moment of inertia I~=I/M^3 and compactness C=M/R, with coefficients depending on the differential-rotation parameter A and the maximum magnetic field Bmax. The authors report an EoS-insensitive scatter with average relative error ~8% and maximum relative error ~50% (Fig. 4), and apply the relation to estimate magnetar spin-down luminosities and the rotational kinetic energy of the GW170817 post-merger remnant. The central claim is that quasi-universality of the I~-C relation extends to highly magnetized, differentially rotating neutron stars.

Significance. If the proposed relation were robust, this would be a useful extension of the Breu-Rezzolla quasi-universal relations into a regime relevant to magnetars and binary-neutron-star merger remnants, and the use of 10 EoSs and fixed-J sequences is a reasonable basis for such a study. The paper also demonstrates a concrete phenomenological path for including differential rotation and magnetic fields in observable estimates. However, the central claim is currently undermined by an internal algebraic inconsistency in Eq. (12), by an A-dependence that is fitted through interpolation across only four A values, and by a validation procedure that re-fits the residuals it later reports as agreement. The maximum relative error of ~50% also tempers the term 'quasi-universal.' With corrections and additional validation, the framework could be valuable, but in its present form the main quantitative claims are not established.

major comments (4)

[Sec. 3.2, Eqs. (6)-(7) and Table 2] Equation (12) as printed is dimensionally inconsistent and does not follow from Eq. (9). With G=c=1, epsilon=Bmax^2 R^4/M^2, so C/sqrt(epsilon) = M^2/(Bmax R^3) and the magnetic contribution in Eq. (9) is Delta I~ = sum_i b_i (C/sqrt(epsilon))^{-i} = sum_i b_i (Bmax R^3/M^2)^i. Rewriting this as coefficients multiplying C^{-i} gives b_i (Bmax R^2/M)^i, not b_i (Bmax R^2/sqrt(GM))^i. Since sqrt(GM)=sqrt(M) has dimensions of length^{1/2} in the G=c=1 units used throughout the paper, the bracket in Eq. (12) is not dimensionless and cannot be added to the dimensionless coefficients a_i and b_i. The printed form overestimates the magnetic term by M^{i/2}; if M is expressed in solar masses, the factors are about 1.18, 1.40, 1.65, and 1.96 for i=1..4. Consequently Eq. (12) cannot be the relation validated in Fig. 4, which was constructed from Eqs. (8)-(9). This error must be corrected and propagated through the applications in Sec. 4.
[Sec. 3.3, Fig. 4] The functions a_i(A) and b_i(A) are obtained by fitting four-parameter cubic polynomials in 1/A to only four values of A (infinity, 10, 8, 6). With four parameters and four data points, the fit is an interpolation, so the reported reduced chi-squared values below 0.02-0.03 do not validate the adopted functional form; in fact the reduced chi-squared has zero degrees of freedom for this interpolation. Moreover, the asymptotic limits of Eq. (7) do not reproduce the uniform-rotation coefficients in Table 2 (for example a1(infinity)=2.0895 versus the tabulated 2.1371, and similar discrepancies for a2-a4 and for the b_i limits in Eq. (10) versus Table 3), contradicting the statement that the relation 'naturally recovers' the uniform rotation limit as A tends to infinity. The A-dependence of Eq. (12) is therefore not established beyond the discrete tabulated values; additional A values or an out-of-sample test are needed.
[Footnote 3 and Sec. 5] The validation in Fig. 4 compares the unified relation with the same data used to construct it. Because Eq. (8) defines the magnetic contribution as the residual after subtracting the rotational fit, and Eq. (9) then fits that residual, the agreement in Fig. 4 is partly guaranteed by construction and does not independently test quasi-universality. The bottom panel of Fig. 4 also reports a maximum relative error of about 50%, which is large for a relation described as quasi-universal; the paper should identify where the worst outliers occur and whether they concentrate at extreme Bmax or at small C. An independent validation, such as leaving out one EoS or one A value and refitting the remaining data, is required to support the claimed 8% average and 50% maximum error estimates.
The paper restricts the numerical analysis to purely toroidal magnetic fields and acknowledges in footnote 3 that such configurations are unstable. Since the proposed unified relation is intended to cover 'highly magnetized' neutron stars, the absence of any test with mixed or poloidal-dominated field configurations is a significant limitation. The additive decomposition Eq. (8) and the polynomial ansatz Eq. (9) are ad hoc and are tested only for toroidal fields; the paper should either provide evidence that the relation holds for mixed-field equilibria or state explicitly that the claimed universality is limited to toroidal-dominated configurations. The discussion in Sec. 5 should be expanded to reflect this caveat in the central claim.

minor comments (5)

[Table 1] The entry 'Relatvistic mean-field' contains a typo; it should read 'Relativistic mean-field.'
[Sec. 5, Discussion] The sentence 'The deviation from the uniform rotation relation increases with with increasing differential rotation' contains a duplicated 'with.'
[Sec. 1, Introduction] The phrase 'may reach upto 10^18 G' should be 'may reach up to 10^18 G.'
[Fig. 2] The color bar in Fig. 2 shows Bmax in the range 0-6 x 10^17 G, while the text states that Bmax is varied over 10^17-10^18 G; please check the range displayed in the figure and state it consistently.
[Sec. 1, Introduction] The sentence 'the current catalog of magnetars comprises of a limited number of observations' is awkward; 'comprises a limited number' or 'consists of a limited number' would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the unified relation is an openly phenomenological fit, and Fig. 4 is an in-sample consistency check rather than an independent prediction.

full rationale

The paper does not present Eq. (12) as derived from first principles; it is explicitly constructed as a fit. Eq. (8) decomposes I-tilde into a rotational part I-tilde_0(C,A), refit from the Breu-Rezzolla form, and a magnetic residual; Eq. (9) fits that residual as a polynomial in C/sqrt(epsilon); Eq. (12) reassembles the two fitted pieces. Because the magnetic contribution is defined as the residual in Eq. (8), the agreement shown in Fig. 4 is in-sample by design, and the paper labels it accordingly: 'To validate the fitted relations, we compute I~(C,A,Bmax) from Eq. (8) for all magnetized and non-magnetized configurations and compare the resulting values with the corresponding original I~.' This is a consistency check of a fit, not a prediction of independent data, so the fitted-input-as-prediction pattern does not apply. The Discussion candidly states that the magnetic treatment 'remains as a phenomenological ansatz and might require additional parametrizations,' and the astrophysical applications are described as 'proof-of-concept' uses of the fitted relation. The Breu-Rezzolla universal relation [29] is external prior work, not a self-citation, and the only author self-citation [45] appears in footnote 3 as a physical plausibility argument for using purely toroidal fields; it is not load-bearing for the algebraic content of Eq. (12). The dimensional inconsistency in Eq. (12) noted by the skeptic is an internal algebraic/correctness issue, not a circularity, and under the hard rules it does not raise the circularity score. No circular step is present.

Assumptions & free parameters 12 free parameters · 7 assumptions · 0 invented entities

The paper's contribution is a set of fitted coefficients, not a first-principles derivation. The physical framework (CFC metric, ideal MHD stress-energy, zero-temperature hadronic EoSs, KEH rotation law) is taken from prior literature, and the polynomial inverse-power ansatz is inherited from Breu and Rezzolla. The genuinely new content is the coefficient dependence on A and Bmax, which is obtained by fitting XNS output, so the free-parameter count is large and the additive ansatz in Eq. (8) is the main ad hoc assumption.

free parameters (12)

a1(A): rotation coefficient in Eq. (7) = 2.0895 + 12.9651/A - 316.8588/A^2 + 1014.7281/A^3
Fitted to I~-C data from XNS for A = infinity, 10, 8, 6; a four-parameter cubic through four points.
a2(A): rotation coefficient in Eq. (7) = -0.4345 - 3.3118/A + 69.6192/A^2 - 171.1361/A^3
Fitted to the same constant-J equilibrium data as a1(A).
a3(A): rotation coefficient in Eq. (7) = 0.1075 + 0.5409/A - 10.9518/A^2 + 23.0102/A^3
Fitted to the same constant-J equilibrium data as a1(A).
a4(A): rotation coefficient in Eq. (7) = -0.0053 - 0.0225/A + 0.3967/A^2 - 0.4320/A^3
Fitted to the same constant-J equilibrium data as a1(A).
b1(A): magnetic coefficient in Eq. (10) = -0.0872 - 0.1829/A + 2.6105/A^2 + 1.5044/A^3
Fitted to the magnetic residual after subtracting the rotational fit.
b2(A): magnetic coefficient in Eq. (10) = -0.0286 - 0.1485/A + 4.4154/A^2 - 17.5769/A^3
Fitted to the magnetic residual after subtracting the rotational fit.
b3(A): magnetic coefficient in Eq. (10) = 0.0007 + 0.0039/A - 0.1084/A^2 + 0.3969/A^3
Fitted to the magnetic residual after subtracting the rotational fit.
b4(A): magnetic coefficient in Eq. (10) = -4.9e-6 - 0.00003/A + 0.0007/A^2 - 0.0023/A^3
Fitted to the magnetic residual after subtracting the rotational fit.
Differential rotation parameter values A = 6, 8, 10, infinity
Only three finite A values plus the uniform limit are used, so the cubic interpolation in 1/A has no independent check between values.
J_const = 0.1968 (code units)
Fixed target angular momentum for every sequence; universality is not tested at other J values.
magnetic index m in barotropic law = 1
Magnetic index in Eq. (4) fixed by hand; only one magnetic field profile family is explored.
eta = Bmax/Bs tuning parameter = 10, 100, 1000
Ad hoc interior-to-surface field ratio in the magnetar application; drives order-of-magnitude changes in the reported spin-down correction.

assumptions (7)

domain assumption CFC/XCFC metric approximation (Eq. 1) is accurate for rotating magnetized neutron stars
Used for all equilibrium models; an approximation rather than exact general relativity, though calibrated in cited works.
domain assumption Ideal magnetized fluid stress-energy with barotropic EoS (Eqs. 2 and 4) describes neutron star matter
Standard GRMHD assumption; zero-temperature EoSs are used.
domain assumption KEH J-constant differential-rotation law (Eq. 3) represents realistic remnants
Chosen because it is common in merger and supernova simulations, but it is not the only possible rotation law.
domain assumption Purely toroidal magnetic configurations approximate physical magnetars
Footnote 3 admits toroidal-only configurations are unstable and no mixed-field models are run; the justification is that toroidal components dominate in the core.
ad hoc to paper Additive decomposition Eq. (8) and magnetic scaling Eq. (9) hold
The magnetic correction is defined as the residual after subtracting the rotational fit and is modeled with an inverse-power form in C/sqrt(epsilon); no derivation of additivity is given.
ad hoc to paper Inverse-power polynomial form of Eq. (5) remains adequate for strong rotation and magnetic fields
Taken from Breu and Rezzolla (2016); Sec. 5 states it is a phenomenological ansatz.
domain assumption Turning-point criterion identifies the stable branch of constant-J sequences
Only stable-branch models are retained, which restricts the domain of the claimed universality.

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Cite this review

Pith. "Pith review of Structural quasi-universality in highly magnetized differentially rotating neutron stars." pith.science (2026). https://pith.science/paper/J27JNVWV

@misc{pith2026260802824,
  author       = {Pith},
  title        = {Pith review of: Structural quasi-universality in highly magnetized differentially rotating neutron stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J27JNVWV}},
  note         = {Machine review of arXiv:2608.02824}
}

read the original abstract

Universal relations among macroscopic properties of neutron stars provide a powerful framework to probe their internal structures while minimizing uncertainties associated with the equation of state (EoS). Although such relations have been extensively studied for uniformly rotating stars, their extension to differentially rotating and strongly magnetized configurations remains largely unexplored. We systematically investigate equilibrium configurations for a wide range of EoSs, rotation profiles, and magnetic field strengths. We establish a generalized quasi-universal relation between the moment of inertia and compactness that remains remarkably insensitive to the underlying EoS across uniformly rotating, differentially rotating, and strongly magnetized configurations. For sequences with fixed angular momentum, the normalized moment of inertia exhibits a quasi-universal dependence on compactness, with deviations primarily due to magnetic field strength and degree of differential rotation. We derive analytic expressions for these dependencies, enabling a unified phenomenological model applicable over a wide range of stellar configurations. As astrophysical applications, we quantify the systematic bias in magnetar luminosity estimates and the rotational kinetic energy of post-merger remnant of GW170817. These results extend quasi-universal relations beyond the standard assumptions of uniform rotation and weak magnetization, providing a robust framework for interpreting observations of highly magnetized and differentially rotating neutron stars, and enabling more reliable constraints on their astrophysical properties.

Figures

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**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 3.** FIG. 3: Contribution on [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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**Figure 6.** Figure 6: FIG. 6: Corner plot showing the correlation among [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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INTRODUCTION Neutron stars (NSs) are the most compact and strongly magnetized stellar remnants currently known, making them unique astrophysical laboratories for probing the behavior of matter under conditions of extreme densi- ties. In the era of multimessenger astronomy [1], a deeper understanding of NS matter is crucial for advancing the fundamental ph...

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METHODOLOGY 2.1. Theoretical Framework In spherical coordinates (t,r,θ,ϕ), the spacetime of a static, axisymmetric compact star can be accurately de- scribed within the conformal flatness condition (CFC) approximation in quasi-isotropic coordinates [30–36]. Under this approximation with the assumptionG=c= 1 withGbeing the Newton gravitational constant and...

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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INTRODUCTION Neutron stars (NSs) are the most compact and strongly magnetized stellar remnants currently known, making them unique astrophysical laboratories for probing the behavior of matter under conditions of extreme densi- ties. In the era of multimessenger astronomy [1], a deeper understanding of NS matter is crucial for advancing the fundamental ph...

work page Pith review arXiv 2026

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METHODOLOGY 2.1. Theoretical Framework In spherical coordinates (t,r,θ,ϕ), the spacetime of a static, axisymmetric compact star can be accurately de- scribed within the conformal flatness condition (CFC) approximation in quasi-isotropic coordinates [30–36]. Under this approximation with the assumptionG=c= 1 withGbeing the Newton gravitational constant and...

work page

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For each configuration,Jis held fixed at a constant valueJ const = 0.1968 by iteratively adjusting the central angular velocity Ωc

THE UNIFIED RELATION In this section, equilibrium sequences of constantJ have been constructed to study the relation between the scaled moment of inertia ˜I=I/M 3 and compactness C=M/Rof NSs, whereMis the mass of the NS andR is its equatorial radius. For each configuration,Jis held fixed at a constant valueJ const = 0.1968 by iteratively adjusting the cen...

work page arXiv 1968

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ASTROPHYSICAL IMPLICATIONS In this section, we show two simple astrophysical ap- plications in magnetars and GWs of the unified relation mentioned in Eq. (8). 4.1. Magnetars We consider a sample of 25 magnetars from the McGill Magnetar Catalog [20] and examine the impact of magnetic-field-induced corrections toIon the inferred spin-down luminosity ( ˙E), ...

work page

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With the help ofxnscode utilizing 10 different EoSs, we have obtained equilibrium configurations of NSs

DISCUSSION In this work, we have investigated the combined ef- fects of differential rotation and magnetic fields on the macroscopic properties of NSs, focusing on two funda- mental observables:IandC. With the help ofxnscode utilizing 10 different EoSs, we have obtained equilibrium configurations of NSs. For sequences of constantJ, we have demonstrated th...

work page 2023

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