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Magnetized spinning quark stars can reach 2.8 solar masses

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2026-08-03 10:34 UTC pith:I4WDTXGI

load-bearing objection New EOS combination, but the headline 2.8 M_sun sits in the regime the authors themselves flag as inaccurate; Eq. (27) also doesn't hold up. the 5 major comments →

arxiv 2601.09529 v1 pith:I4WDTXGI submitted 2026-01-14 astro-ph.HE

Strange quark star I: the maximum gravitational mass and deformation of magnetized spinning model

Fatemeh Kayanikhoo , Mateusz Kapusta , Miljenko \v{C}emelji\'c , Leszek Zdunik This is my paper
classification astro-ph.HE
keywords strange quark starsMIT bag modeldensity-dependent bag constantLandau quantizationmagnetized rotating starsmaximum gravitational massdeformation parameterGW190814
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that strange quark stars built from a density-dependent bag constant and Landau-quantized quarks can be considerably heavier than earlier fixed-bag models allowed, with maximum gravitational masses climbing from 2.35 M⊙ at rest to 2.8 M⊙ at 1200 Hz and 5×10^17 G. It also finds that such fast, magnetized stars become strongly oblate, with an equatorial-to-polar radius ratio up to 1.55. If these models hold, strange quark stars could explain heavy compact objects like the black widow pulsar J0952-0607 and the 2.5–2.67 M⊙ secondary in GW190814. The paper packages its results into fitted formulas for maximum mass and deformation as functions of magnetic moment and spin frequency.

Core claim

Using the MIT bag model with a bag constant that falls with baryon density (from 400 to about 9 MeV/fm³) and including Landau quantization at fields up to 5×10^17 G, the authors construct equations of state for strange quark matter and solve the axisymmetric Einstein equations for rotating, magnetized stars. In the non-rotating, non-magnetized case the maximum gravitational mass is 2.35 M⊙; rotation alone raises it to 2.73 M⊙ at 1200 Hz, and the strongest central field adds roughly another 0.07 M⊙, giving 2.8 M⊙ in the combined model. The maximum deformation parameter a = Req/Rpol reaches 1.55 in the same fast-rotating, strongly magnetized configuration. Magnetic energy contributes less than

What carries the argument

The central mechanism is the density-dependent MIT bag equation of state: a bag 'constant' that starts near 400 MeV/fm³ and drops to about 9 MeV/fm³ at high baryon density, which stiffens matter at high density and raises maximum masses relative to a fixed bag constant. Landau quantization enters through the Fermi relations, discretizing transverse quark motion and determining how pressure responds to fields up to 5×10^17 G. These EOS inputs feed a spectral-method general-relativistic solver for uniformly rotating magnetized stars, and the results are summarized by fitted formulas in which maximum mass and deformation each factor into separate magnetic-moment and spin-frequency terms.

Load-bearing premise

The headline numbers rest on trust that the general-relativistic magnetized-star solver computes the strongly magnetized, rapidly rotating regime accurately, a regime the paper itself says is a limitation with virial errors near 1e-2.

What would settle it

Recompute the same equation of state at B_c = 5×10^17 G and f = 1200 Hz with an independent spectral or finite-volume general-relativistic magnetized-star code and verify the virial identity to better than 1e-4; if the resulting maximum mass falls below about 2.5 M⊙ or the deformation parameter fails the expected stability criteria, the paper's headline claims would be contradicted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, strange quark stars can have gravitational masses above 2.3 M⊙, making them viable explanations for PSR J0952-0607 and the GW190814 secondary.
  • Rotation raises the maximum mass from 2.35 M⊙ to 2.73 M⊙ at 1200 Hz, with the magnetic field contributing an additional ~0.07 M⊙, so spin acts as a mass-raising mechanism, not just a shape-changing one.
  • At 1200 Hz, only sufficiently massive configurations exist in the model; low-mass strongly magnetized, rapidly rotating strange quark stars are absent, which the companion paper discusses in terms of a minimum-mass limit.
  • The fitted mass formula (1+aµ²)(1+bf^c), with c≈2.7, gives a simple way to estimate maximum mass from observable spin and magnetic-moment data.
  • Binding energy is approximately a linear function of compactness, E_BE/M_g ≈ −β, across all computed field strengths and rotational frequencies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The factorized form of the fitted formulas suggests spin and magnetic field act nearly independently on maximum mass and shape; if that factorization holds for other equations of state, it could serve as a cheap surrogate for full numerical stellar-structure calculations.
  • The paper's own flagged limitation in the strongly magnetized, rapidly rotating regime — virial accuracy near 1e-2 — sits exactly at the parameters producing the headline 2.8 M⊙ and 1.55 values, so a convergence study reaching 1e-4 would establish whether those numbers move.
  • The predicted low-mass cutoff at 1200 Hz, if real, implies that fast-spinning quark-star pulsars should be heavy; future surveys of millisecond pulsars could test this population-level consequence.
  • Extrapolating the fitted formulas slightly beyond 1200 Hz suggests masses approaching 3 M⊙ might be allowed, but the practical ceiling depends on the Keplerian breakup limit taken up in the companion paper.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper constructs a strange quark star EOS from a density-dependent MIT bag model with Landau quantization of quarks in a strong magnetic field (up to 5×10^17 G). Using the LORENE spectral solver (Et_magnetisation class), it computes sequences of uniformly rotating, poloidally magnetized equilibrium models at f=0, 400, 800, and 1200 Hz, with central fields up to ~5×10^17 G. From these sequences it reports the maximum gravitational mass and the deformation parameter a=R_eq/R_pol, and fits functions M_max(μ,f) and a(μ,f). The headline results are M_max=2.80 M_sun and a=1.55 at f=1200 Hz and B_c≈5×10^17 G. The paper compares these masses with PSR J0952-0607 (2.35 M_sun) and GW190814 (2.5–2.67 M_sun), and studies binding energy, external energy, and compactness, claiming a linear E_BE/M_g = −β relation.

Significance. If the numerical results are correct, the paper would show that a specific strange quark matter equation of state, when combined with strong magnetic field and rapid rotation, can produce compact objects exceeding 2.5 M_sun, potentially explaining massive compact-object observations without invoking exotic hadronic matter. The work uses a recognized spectral code (LORENE), provides a tabulated maximum-mass data set (Table 1), and offers analytic fitting functions for M_max and a. However, the decisive maximum mass and deformation values come from the one corner of parameter space (B_c~5×10^17 G, f=1200 Hz) where the authors report only ~10^-2 virial accuracy and explicitly disclaim accurate critical configurations. The absence of a convergence study and the presence of a jump in R_circ and a in the last row of Table 1 mean the headline values are not established. The EOS specification also omits the up/down quark masses needed for Landau quantization, and Eq. (27) is numerically inconsistent. The paper's significance is therefore conditional on addressing these points.

major comments (5)
  1. [§4 and §6; Table 1] The headline values (M_g=2.80 M_sun, a=1.55) are computed in the same regime for which the authors report virial accuracy only ≈10^-2 and state that a limitation of the current model lies in accurately calculating critical configurations of strongly magnetized, rapidly rotating stars. The last row of Table 1 (B_c=4.97×10^17 G, f=1200 Hz) shows a simultaneous jump in R_circ (14.21→14.6 km) and a (1.46→1.55) with M_g unchanged, unlike the smoother trends in earlier rows. No resolution or convergence study is presented. Because the central claims rest on this corner of parameter space, a convergence study (e.g., number of spectral domains/grid points) and quantitative error bars for M_g, R_circ, and a are required.
  2. [§5.5, Eq. (27)] The relation E_BE/M_g = −β with β=M_g/R_circ is dimensionally inconsistent as written (E_BE/M_g is dimensionless, while M_g/R_circ in M_sun/km is not) and is contradicted by Table 1. For the nonmagnetized nonrotating model, β=2.35/11.92=0.197 M_sun/km while |E_BE|/M_g=(2.92−2.35)/2.35=0.243; for the last row, β=0.192 M_sun/km while |E_BE|/M_g≈0.225. If β is meant to be the dimensionless compactness GM/(R c^2), the numerical values must be recomputed and the relation re-fit.
  3. [§2, Eqs. (2)–(8)] The Landau quantization uses B_C=m_i^2 c^3/(q_iℏ). The paper specifies only m_s=150 MeV; the up and down quark masses are not given. If u and d are taken massless (common in MIT bag EOS), then B_C=0 and the expressions for ϵ_i and ν_max are singular. If nonzero masses are used, they must be stated explicitly, since they enter every thermodynamic quantity. Please specify all quark masses and show that the low-mass limit is regular.
  4. [§5.2] The claim that models with M_g≥2.5 M_sun are compatible with GW190814 is not controlled for spin. In Table 1, masses above 2.5 M_sun are reached only at f≥800 Hz and B_c≳4×10^17 G. The GW190814 secondary is not known to be a rapidly rotating, strongly magnetized object; the paper does not discuss the spin constraints on the source. The PSR J0952-0607 comparison is more appropriate because that object rotates near 700 Hz, but the GW190814 compatibility needs either a justification that such parameters are relevant or a caveat.
  5. [§5.1] The stability criterion used to identify the maximum stable configuration is stated as dM/dR_J>0. The standard turning-point criterion for secular stability in rotating stars is (∂M/∂ρ_c)_J=0 along a constant-J sequence. The equivalence to dM/dR_J>0 requires that R vary monotonically with ρ_c, which is not shown and is not guaranteed for all EOS and rotation rates. Please justify the criterion or use the standard one; this directly determines the M_max values reported in Table 1.
minor comments (6)
  1. [§5.2] The rotation period of PSR J0952-0607 is 1.4 ms, not 1.4 s; also the name is hyphenated as PSR J0952-0607, not PSR J09520607.
  2. [§1] Reference '(Chakrabarty et al. 1989,?)' contains a stray question mark; the full bibliographic entry is incomplete.
  3. [§5.4, Eq. (24)] The surface integral is written as '∫ dV/da (n · φ∇φ)', which is unreadable; it should be a surface integral ∫ dA n·(φ∇φ).
  4. [Table 1] Please verify the f=1200 Hz, B_c=1.04×10^17 G row: R_circ jumps from 13.08 to 13.84 km while M_g and a are unchanged from the B=0 row; this may be a typo or a sign of a nearby convergence issue.
  5. [§5.1] The fit in Eq. (22) uses the magnetic moment μ, but the text introduces it as a function of magnetic field strength; clarify that μ is the computed magnetic moment of each model.
  6. [Fig. 1] The EOS curves for the four magnetic field strengths overlap and are barely distinguishable; the legend is not sufficient. Use distinct line styles or an inset.

Circularity Check

0 steps flagged

No significant circularity: the headline masses/deformations are genuine LORENE outputs with literature-based EOS inputs; fitted functions are explicit summaries, not predictions.

full rationale

The derivation is self-contained. The EOS is built from the MIT bag model with density-dependent bag constant (Eq. 9) and Landau-quantized Fermi relations (Eqs. 2-8); all parameters (ms=150 MeV, rho0=0.17 fm^-3, alpha=0.17, B0=400 MeV/fm^3, Binf=8.99 MeV/fm^3) are taken from prior literature or from matching quark/hadronic energy density, not from the astrophysical masses later compared with observations. The stellar-structure results are obtained by solving the Einstein-Maxwell system with LORENE's Et_magnetisation class; the maximum masses, radii, and deformation parameters in Table 1 are numerical outputs, not definitions of the inputs. The fitted functions in Eqs. (22) and (23) are explicitly fits to the computed data ('We fit a function on the data...'), and the paper presents them as characterizations rather than independent predictions, so no fitted quantity is relabeled as a prediction. The comparison with PSR J0952-0607, GW190814, and SGR J1745-2900 is ex post and does not feed back into the EOS. The only self-citations (Gondek-Rosinska et al. 2000, Haensel et al. 2009) are background context and are not used to justify the headline results. The paper's own caveat--'A limitation of our current model lies in accurately calculating the critical configurations of strongly magnetized, rapidly rotating stars'--plus the reported ~1e-2 virial accuracy in that regime is a numerical reliability concern for the 2.8 Msun/a=1.55 endpoint, not a circularity. No step reduces to its own input.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The model is not self-contained: it inherits the MIT bag parameterization (B0, B∞, α, ρ0), the strange quark mass, and LORENE's numerical solver from prior work. The paper's own free parameters are the six coefficients fitted to its numerically generated M_max and deformation data. No new particles, forces, or dimensions are introduced.

free parameters (4)
  • MIT bag parameters (B0, B∞, α, ρ0) = B0 = 400 MeV/fm^3, B∞ = 8.99 MeV/fm^3, α = 0.17, ρ0 = 0.17 fm^-3
    Density-dependent bag constant in Eq. (9), taken from Burgio et al. (2002)/CERN-SPS; the high mass predictions depend directly on the low B∞ value.
  • strange quark mass m_s = 150 MeV
    Set in §2; affects the Landau quantization and the EOS stiffness.
  • maximum-mass fit coefficients a, b, c = a = 2.34e-2 A^-2 m^-4, b = 6.88e-10 s^2.72, c = 2.72
    Fitted to the computed M_max(μ,f) grid, Eq. (22); this is a fit, not a prediction.
  • deformation fit coefficients ã, b̃, č = ã = 8.03e-2 A^-2 m^-4, b̃ = 2.22e-10 s^2.66, č = 2.66
    Fitted to the computed shape data, Eq. (23); not a derivation.
axioms (6)
  • domain assumption MIT bag model provides a valid equation of state for strange quark matter
    Underlies all results; introduced in §2 with the density-dependent bag constant.
  • standard math Landau quantization formula for charged quarks in strong magnetic fields (Eqs. 2–8)
    Assumes free quarks with only Landau-level modification; neglects anomalous magnetic moments, pairing, and other many-body effects.
  • domain assumption Axisymmetric, uniformly rotating equilibrium with a poloidal magnetic field (LORENE Et_magnetisation)
    Restricts the configuration space; stated in §3–§4.
  • domain assumption Turning-point stability criterion sign(dM/dR)_J > 0
    Used in §5.1 to identify maximum stable configurations; this is an approximate secular stability criterion.
  • domain assumption Electron contribution can be neglected
    §2 states the electron fraction is of order 1e-3, so electrons are dropped from the EOS.
  • domain assumption External magnetic energy can be computed in flat space with a vacuum dipole potential
    §5.4 explicitly neglects general-relativistic effects on the external magnetic field.

pith-pipeline@v1.3.0-alltime-deepseek · 13555 in / 16175 out tokens · 161488 ms · 2026-08-03T10:34:48.723085+00:00 · methodology

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read the original abstract

We investigate the structural parameters of strange quark stars (SQS) under the influence of strong magnetic fields and varying rotational frequencies. The equation of state is computed using the MIT bag model with a density-dependent bag constant and considering the Landau quantization effect regarding the strong magnetic fields up to $5\times10^{17}\,$G in the interior of SQS. Employing the LORENE library, we calculate the structural parameters under different magnetic field strengths and rotational frequencies. Our models are compared in terms of maximum gravitational mass, deformation parameter, binding energy, and compactness. Our equation of state model demonstrates that the gravitational masses are higher than those computed using a MIT bag model with a fixed bag constant. We find the gravitational masses beyond $2.3 \,M_\odot$, which are compatible with the masses of observed compact objects, such as the ``black widow'' pulsar \emph{PSR J0952-0607}, and the \emph{GW190814} event detected by the LIGO/Virgo collaboration. The deformation parameter and maximum gravitational mass of SQS are characterized by fitted functions accounting for variations in both magnetic field strength and rotational frequency. We find the maximum deformation parameter of 1.55 and the maximum gravitational mass of $2.8\, M_\odot$ in the fast-rotating strongly magnetized model.

Figures

Figures reproduced from arXiv: 2601.09529 by Fatemeh Kayanikhoo, Leszek Zdunik, Mateusz Kapusta, Miljenko \v{C}emelji\'c.

Figure 1
Figure 1. Figure 1: Pressure as a function of the energy density of SQM in the presence of different magnetic field strengths. The energy density corresponding to the maximum gravitational mass is pointed with the red cross. The curves with four magnetic fields indicated in the legend overlap. sionless magnetic field, where BC = m2 i c 3/qiℏ, with qi the charge of quark i. The number density of quarks is obtained from ρ = νXm… view at source ↗
Figure 2
Figure 2. Figure 2: Gravitational mass Mg versus circumferential radius Rcirc for different rotational frequencies in each panel, with the colors indicating the value of the central magnetic field, as indicated in the color bar. at constant angular momentum J. An example is illus￾trated in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Gravitational mass Mg versus circumferential radius Rcirc with different angular momentum (J = 1.13, 2.26, 3.39 [GM2 ⊙/c]) in each panel with the colors indicating the value of the central magnetic field [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Gravitational mass Mg as a function of central energy density in the unit of [ρnucc 2 ] with different angular momentum (J = 1.13, 2.26, 3.39 [GM2 ⊙/c])in each panel with the colors indicating the value of the central magnetic field [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Gravitational mass Mmax g as a function of the magnetic moment µ at different rotational frequencies f. Each color represents one rotational frequency. The dotted lines show the fitted function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The stellar shape of SQS in different computed models. The color of the ellipses indicates the magnitudes of the central magnetic field. Straight lines which are at the same positions in all panels indicate the polar radius of the non-magnetized configuration in each rotational frequency [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Deformation parameter a versus magnetic moment µ in different rotational frequencies for the maximum stable configurations of SQS in each computed model [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

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