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

This study utilizes the MIT bag model with a density-dependent bag constant and Landau quantization to demonstrate that strongly magnetized, rapidly rotating strange quark stars can achieve gravitational masses up to $2.8\,M_\odot$ and deformation parameters of 1.55, thereby providing theoretical models consistent with observed massive compact objects like PSR J0952-0607 and GW190814.

Original authors: Fatemeh Kayanikhoo, Mateusz Kapusta, Miljenko Čemeljić, Leszek Zdunik

Published 2026-06-18

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CC BY 4.0

Original authors: Fatemeh Kayanikhoo, Mateusz Kapusta, Miljenko Čemeljić, Leszek Zdunik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: What is a "Strange" Star?

Imagine a star not made of the usual gas and plasma, but squashed so hard that its atoms break apart. Inside these "Strange Quark Stars" (SQS), the building blocks of matter (protons and neutrons) dissolve into a soup of their smaller parts: quarks. Specifically, these stars are made of "up," "down," and "strange" quarks.

Think of a normal star like a fluffy cloud of cotton candy. A strange quark star is like that same amount of sugar, but compressed into a single, incredibly dense sugar cube. The paper asks: How heavy can this sugar cube get before it collapses? And what happens if you spin it really fast or hit it with a giant magnet?

The Ingredients: The "Bag" and the "Spin"

To figure this out, the scientists used a recipe called the MIT Bag Model.

The Bag: Imagine the quarks are trapped inside a rubber balloon (the "bag"). The pressure of the balloon keeps them from flying apart. In this study, the scientists made the rubber of the balloon change its stiffness depending on how crowded the quarks are inside.
The Magnet: These stars are often surrounded by magnetic fields so strong they would wipe a credit card clean from light-years away. The paper simulates fields up to $5 \times 10^{17}$ Gauss (for comparison, a fridge magnet is about 50 Gauss).
The Spin: The stars are also spinning incredibly fast, up to 1,200 times per second (like a blender on high speed).

The Experiment: Building Digital Stars

The researchers didn't build a real star (which is impossible). Instead, they built a digital simulation using a super-computer tool called LORENE.

They created thousands of virtual stars with different settings:

No spin, no magnet: The baseline.
Spinning fast: How does the centrifugal force (the force that pushes you outward on a merry-go-round) change the star?
Super-magnetic: How does the magnetic field squeeze or stretch the star?
Both: The ultimate test.

Key Findings: The "Super-Heavy" Stars

1. How Heavy Can They Get?
In the simplest model (no spin, no magnet), the star can weigh about 2.35 times the mass of our Sun.

The Twist: When you add a strong magnetic field and spin it fast, the star gets heavier.
The Result: In the most extreme scenario (spinning at 1,200 Hz with a massive magnetic field), the star can hold up to 2.8 times the Sun's mass.

Why does this matter?
Astronomers have found real objects in space that are very heavy, like the "Black Widow" pulsar (about 2.35 solar masses) and a mysterious object from a collision called GW190814 (between 2.5 and 2.67 solar masses).

The Paper's Claim: Their "Strange Quark Star" model is heavy enough to explain these real-life cosmic mysteries. It suggests that these heavy objects might actually be made of quark soup, not just normal neutron matter.

2. The Shape Shift: The "Oblate" Effect
If you spin a ball of dough, it flattens out at the poles and bulges at the equator.

The Analogy: Imagine a spinning pizza dough. The faster it spins, the wider it gets.
The Finding: The magnetic field makes the star slightly smaller (squashing it), but the spin makes it very wide.
The Record: The most extreme model (fast spin + strong magnet) is the most "flattened" star they found. It is 1.55 times wider at the equator than it is tall from pole to pole. It's like a giant, cosmic flying saucer.

3. The Energy Balance
The team calculated the total energy of these stars.

They found that the energy stored in the magnetic field is actually quite small compared to the total energy of the star (less than 1%).
However, the total energy of these stars is massive—about $10^{54}$ ergs. To put that in perspective, a typical supernova explosion (a star dying) releases about $10^{51}$ ergs. These strange stars hold enough energy to power thousands of supernovas, though the paper notes this is just a calculation of their internal state, not a prediction of an explosion.

The Bottom Line

This paper is a theoretical "stress test" for Strange Quark Stars. By using a specific recipe for how quarks behave under extreme pressure, and adding the effects of super-magnets and high-speed spins, the authors found that:

These stars can be heavier than previously thought (up to 2.8 solar masses).
This extra weight helps explain some of the heaviest, strangest objects astronomers have recently detected.
These stars can become very flat (like a pancake) when they spin fast and are magnetized.

The authors conclude that their model is a strong candidate for explaining the nature of these exotic, heavy, and fast-spinning cosmic objects.

Technical Summary: Strange Quark Star I: The Maximum Gravitational Mass and Deformation of Magnetized Spinning Model

Problem Statement
The paper addresses the structural parameters of Strange Quark Stars (SQS) under the extreme conditions found in compact astrophysical objects: strong magnetic fields (up to $5 \times 10^{17}$ G) and rapid rotation (frequencies up to 1200 Hz). While the existence of SQS—hypothetical compact objects composed entirely of strange quark matter (SQM)—remains unconfirmed, recent observations of massive compact objects (e.g., PSR J0952-0607 and the GW190814 event) challenge standard neutron star models. The authors aim to determine if an SQS model, incorporating density-dependent equations of state (EOS), strong magnetic fields, and rotation, can support gravitational masses consistent with these observations while characterizing the resulting stellar deformation and energetics.

Methodology
The study employs a multi-faceted theoretical and numerical approach:

Equation of State (EOS): The authors utilize the MIT bag model with a density-dependent bag constant ( $B_{bag}(\rho)$ ) defined by a Gaussian relation. To account for the strong internal magnetic fields, the Fermi relations for quarks (up, down, and strange) are modified to include Landau quantization effects. The model assumes a zero-temperature, zero-electron fraction approximation, neglecting electron contributions as they are of the order of $10^{-3}$ .
Stellar Structure Equations: The Einstein field equations are solved in an axisymmetric spacetime using the 3+1 formalism. The energy-momentum tensor includes contributions from the perfect fluid, magnetization, and the pure magnetic field. This results in a system of four elliptic partial differential equations (PDEs) governing the metric functions.
Numerical Implementation: The authors use the LORENE library, a spectral method-based C++ code, to solve the PDEs. The computational domain is divided into three regions: the stellar interior, a thin shell near the surface, and an exterior extending to infinity.
Parameter Space: Calculations were performed for rotational frequencies of 0, 400, 800, and 1200 Hz, and central magnetic field strengths ranging from 0 to $5 \times 10^{17}$ G. For each configuration, 51 stellar models were computed by varying the central enthalpy from $0.01c^2$ to $0.51c^2$ .
Validation: The accuracy of the numerical solutions was verified using the 3-dimensional and 2-dimensional general relativistic virial identities (GRV3 and GRV2), yielding errors of $\approx 10^{-5}$ for non-magnetized/non-rotating models and $\approx 10^{-2}$ for magnetized/fast-rotating models.

Key Contributions and Results

Maximum Gravitational Mass ( $M_g^{max}$ ):
- The model predicts a maximum gravitational mass of 2.35 $M_\odot$ for a non-magnetized, non-rotating SQS.
- Increasing the magnetic field strength (up to $5 \times 10^{17}$ G) in a non-rotating model slightly increases $M_g^{max}$ to 2.37 $M_\odot$ .
- Rotation significantly enhances the maximum mass. In the fastest rotating model (1200 Hz) with the strongest magnetic field ( $5 \times 10^{17}$ G), the maximum gravitational mass reaches 2.80 $M_\odot$ .
- The authors provide a fitted function relating $M_g^{max}$ to magnetic moment ( $\mu$ ) and rotational frequency ( $f$ ): $M_g^{max}(\mu, f)/M_g^{max}(0,0) \simeq (1 + a\mu^2)(1 + bf^c)$ .
Deformation Parameter ( $a$ ):
- The deformation parameter is defined as the ratio of the equatorial radius to the polar radius ( $a = R_{eq}/R_{pol}$ ).
- The study finds that rotation is the dominant factor in deformation, while the magnetic field has a minor effect on oblateness.
- The maximum deformation parameter found is 1.55, occurring in the model with $f = 1200$ Hz and $B_c = 5 \times 10^{17}$ G.
- A fitted function for deformation is also provided: $a(\mu, f)/a(0,0) \simeq (1 + \tilde{a}\mu^2)(1 + \tilde{b}f^{\tilde{c}})$ .
Energetics and Stability:
- Binding Energy: The total binding energy per baryon ( $|E_{BE}|/A$ ) ranges between 171 MeV and 184 MeV, confirming the stability of the SQS models. The ratio of binding energy to gravitational mass is approximately 0.24, showing a linear relationship with compactness ( $\beta = M_g/R_{circ}$ ).
- Total Energy: The total energy of the SQS is estimated at $\approx 5.5 \times 10^{47}$ J ( $\approx 10^{54}$ ergs). The contribution of external magnetic energy to the total energy is less than 1%.
- Stability Criterion: Stability against axisymmetric perturbations was assessed by ensuring $(dM/dR)_J > 0$ .
Comparison with Observations:
- The model's predicted masses are compatible with observed compact objects. Specifically, the non-rotating, non-magnetized mass (2.35 $M_\odot$ ) aligns with PSR J0952-0607.
- The higher mass limits (up to 2.80 $M_\odot$ ) in rotating/magnetized models are consistent with the mass range of the secondary component in the GW190814 event (2.5–2.67 $M_\odot$ ).
- The models also accommodate the mass and radius estimates for the magnetar SGR J1745–2900 and other massive pulsars like PSR J1614-2230 and PSR J0348+0432.

Significance and Claims
The paper claims that the proposed SQS model, utilizing a density-dependent MIT bag model with Landau quantization, successfully reproduces the high gravitational masses observed in recent astrophysical events without requiring exotic physics beyond the standard quark matter framework. The authors emphasize that while magnetic fields below $10^{19}$ G do not significantly alter the stiffness of the EOS (as noted in previous studies by Chatterjee et al.), they, combined with rotation, significantly alter the structural parameters (mass and shape) of the star.

The study establishes that rapidly rotating, strongly magnetized SQSs can support masses up to 2.8 $M_\odot$ , offering a potential explanation for the "mass gap" objects detected by LIGO/Virgo and heavy pulsars. The authors explicitly state that the minimum gravitational mass limits and Keplerian configurations for these models are explored in a companion paper, and they acknowledge that accurately calculating critical configurations for strongly magnetized, rapidly rotating stars remains a limitation for future refinement.

The Big Picture: What is a "Strange" Star?

The Ingredients: The "Bag" and the "Spin"

The Experiment: Building Digital Stars

Key Findings: The "Super-Heavy" Stars

The Bottom Line

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