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Strange quark star II: the minimal and maximal gravitational mass and the Keplerian configuration

Using the MIT bag model with a density-dependent bag constant and the LORENE library, this study demonstrates that rapidly rotating non-magnetized strange quark stars must possess a minimum mass to sustain high rotational frequencies, with their Keplerian frequency scaling linearly with gravitational mass and yielding mass limits consistent with observed pulsar data.

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

Published 2026-06-18

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CC BY 4.0

Original authors: Fatemeh Kayanikhoo, Mateusz Kapusta, Miljenko Čemeljić, Wlodek Kluzniak, 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

Imagine the universe as a giant kitchen. Usually, when you cook matter, you get atoms—tiny solar systems of protons, neutrons, and electrons. But deep inside the cores of the densest objects in the universe, like neutron stars, the pressure is so immense that it's like crushing a car into the size of a sugar cube. At this point, the "glue" holding the atoms together breaks, and the ingredients (quarks) spill out to form a new, super-dense soup called Strange Quark Matter.

If a star is made entirely of this soup, it's called a Strange Quark Star (SQS). This paper is a recipe book for these stars, specifically looking at how heavy they can be, how light they can be, and how fast they can spin before they fall apart.

Here is the breakdown of their findings using simple analogies:

1. The "Bag" of Quarks

To understand these stars, the scientists used a model called the MIT Bag Model.

The Old Way: Imagine a bag of marbles where the bag itself is rigid and unchanging. This is the "fixed bag constant." It's a simple rule, but it doesn't quite match the heavy stars we see in the sky today.
The New Way (This Paper): The authors used a density-dependent bag. Think of this bag as being made of a smart, stretchy material. As you squeeze the bag tighter (higher density), the material changes its properties. This "smart bag" allows the star to hold more weight without collapsing, which matches the observations of the heaviest pulsars we've found.

2. The Spin Limit: The "Spinning Pizza"

The paper looks at what happens when these stars spin incredibly fast—thousands of times per second.

The Analogy: Imagine a pizza chef spinning a ball of dough. If they spin it too fast, the dough flies off the edges.
The Discovery: The scientists found that for a strange quark star to spin this fast (between 1,100 and 1,300 times a second), it cannot be too light.
- If the star is too light, the centrifugal force (the force pushing things outward) is too strong for its own gravity to hold it together. It would fly apart like that pizza dough.
- Therefore, there is a minimum weight required to survive a high-speed spin. In their model, a star spinning at 1,300 Hz needs to be at least about 2 times the mass of our Sun to stay intact.

3. The Heavy Limit: The "Super-Stack"

The Analogy: Think of stacking bricks. Usually, if you stack them too high, the bottom ones crush.
The Discovery: Because their "smart bag" model is stiffer (stronger) at high densities, these stars can support much more weight than older models predicted.
- Old models said the limit was about 1.4 to 1.9 times the Sun's mass.
- This new model says these stars can weigh up to 2.35 times the Sun's mass even when they aren't spinning, and up to 2.87 times when they are spinning fast.
- This is great news for the model because it explains the existence of the heaviest pulsars we've actually detected, which weigh around 2.35 times the Sun.

4. The "Speed vs. Weight" Rule

The paper found a neat, straight-line relationship between how fast the star spins and how heavy it must be.

The Analogy: It's like a speed limit sign that changes based on the weight of your car. The heavier the car, the faster it can safely go on a specific curve.
The Result: They found that for every extra bit of mass the star has, the maximum safe spinning speed increases by a predictable amount. This linear relationship helps astronomers guess the mass of a star just by measuring how fast it spins.

5. Does it Match Reality?

The scientists compared their "smart bag" recipe against real data from telescopes:

The Heavyweights: It perfectly explains the heaviest pulsars found (like PSR J0952-0607).
The Lightweights: It also fits the data for the lightest compact object found (HESS J1731-347), which might be a strange quark star with a very exotic recipe.
The "Goldilocks" Zone: Their model suggests that strange quark stars can exist across a wide range of sizes, from the lightest detected objects to the heaviest ones, provided they spin fast enough if they are on the lighter side.

Summary

In short, this paper argues that if strange quark stars exist, they are likely made of a "smart" type of matter that gets stronger under pressure. This allows them to be much heavier than previously thought and requires them to have a minimum weight if they want to spin at breakneck speeds without flying apart. Their calculations align perfectly with the heaviest and lightest cosmic objects we have observed so far.

Technical Summary: Strange Quark Star II: The Minimal and Maximal Gravitational Mass and the Keplerian Configuration

Problem Statement
The existence of strange quark stars (SQS) remains a theoretical possibility supported by the hypothesis that strange quark matter (SQM) may be the true ground state of matter, potentially more stable than nuclear matter. While previous studies have explored the structural parameters of SQSs, often utilizing the MIT bag model with a fixed bag constant ( $B_{bag}$ ), there is a need to refine these models to align with recent observational data. Specifically, recent detections of both very heavy pulsars (exceeding $2 M_\odot$ ) and very light compact objects (such as HESS J1731-347 with $M_g \approx 0.77 M_\odot$ ) challenge existing equations of state (EOS). This paper investigates the minimal and maximal gravitational masses and the Keplerian (mass-shedding) configurations of rapidly rotating SQSs using a density-dependent MIT bag model, aiming to determine if such a model can simultaneously accommodate the full range of observed compact object masses.

Methodology
The authors employ the MIT bag model with a density-dependent bag constant, $B_{bag}(\rho)$ , to construct the EOS. This approach, based on experimental results from CERN regarding quark-gluon plasma formation, defines the bag constant as:
$B_{bag}(\rho) = B_\infty + (B_0 - B_\infty)e^{-\alpha(\rho/\rho_0)^2}$
where $\rho$ is the baryon number density. This model yields a variable slope ( $a$ ) in the pressure-energy density relation, ranging from $a \approx 1$ at low densities to $a \approx 0.26$ at high densities, contrasting with the fixed slope of traditional models.

To compute the structural parameters of rapidly rotating SQSs, the authors utilize the LORENE library, which solves the Einstein field equations in a 3+1 formalism for stationary, axisymmetric spacetimes using spectral methods. The study covers rotational frequencies in the range of 1100–1300 Hz. For each frequency, the central enthalpy is varied to map the mass-radius sequences. The authors specifically analyze the mass-shedding limit (Keplerian configuration), where the equatorial surface velocity matches the orbital velocity of a test particle.

Key Contributions and Results

EOS Comparison and Static Limits: The density-dependent bag model ( $B_{bag}(\rho)$ ) predicts a maximum static gravitational mass of $2.35 M_\odot$ , significantly higher than the $\approx 1.96 M_\odot$ predicted by the fixed bag constant model ( $B_{bag}^{fixed} = 60 \text{ MeV/fm}^3$ ) and the $\approx 1.3-1.4 M_\odot$ limits of earlier SS1/SS2 models. This higher mass limit is necessary to be consistent with the heaviest observed pulsars.
Rotational Mass Limits: The study finds that for rapidly rotating SQSs, the maximum gravitational mass ( $M_g^{max}$ ) increases with rotational frequency, rising from $2.55 M_\odot$ at 1100 Hz to $2.87 M_\odot$ at 1300 Hz. Conversely, the minimum gravitational mass ( $M_g^{min}$ ) required to sustain a given rotational frequency also increases with frequency, ranging from $0.25 M_\odot$ at 1100 Hz to $2.15 M_\odot$ at 1300 Hz.
Keplerian Frequency and Mass Relationship: A primary finding is the linear relationship between the Keplerian frequency ( $f_{Kep}$ ) and the gravitational mass at the mass-shedding limit (and similarly for the minimum mass). The authors derive a slope of approximately $0.08 \text{ kHz}/M_\odot$ ( $f \approx 0.078 M_g + 1.13$ ). This linear dependence holds for frequencies $\ge 1150$ Hz, where a distinct minimum mass configuration and mass-shedding limit emerge.
Validation of Approximate Formulas: The authors test the approximate formula for Keplerian frequency proposed by Haensel et al. (2009). They find that for configurations with $M_g \ge 1.5 M_\odot$ , the formula agrees with their numerical solutions to within a fraction of a percent. For lower masses ( $M_g \le 1.4 M_\odot$ ), the formula slightly overestimates the frequency by less than 2.55%.
Observational Consistency: The model successfully accounts for the mass limits of both the heaviest and lightest observed compact objects. It supports the existence of pulsars with $M_g \gtrsim 2 M_\odot$ (e.g., PSR J0952-0607) and is compatible with the low mass of HESS J1731-347, provided the object is rotating rapidly enough to satisfy the minimum mass constraint for its frequency.

Significance and Claims
The paper claims that the density-dependent MIT bag model provides a robust theoretical framework for strange quark stars that resolves tensions between previous models and modern observational data. By demonstrating that SQSs can support masses up to $\approx 2.35 M_\odot$ (static) and higher when rotating, the model validates the potential existence of SQSs as the progenitors of the most massive pulsars. Furthermore, the identification of a minimum mass threshold for high-frequency rotation suggests a specific structural characteristic of bare strange stars: they cannot sustain arbitrarily high spin rates unless they possess a sufficient gravitational mass to maintain hydrostatic equilibrium against centrifugal forces.

The authors note limitations, specifically that numerical accuracy restricts the determination of minimum mass configurations for frequencies below 1175 Hz and that calculations become unstable above 1275 Hz, requiring extrapolation. However, within the computed range, the results align with observational constraints for both the heaviest and lightest known compact objects, reinforcing the viability of the SQS hypothesis under a density-dependent equation of state.