Maximum Mass Limit of Strange Stars

Updated 17 August 2025

Strange stars are compact stellar objects composed solely of deconfined up, down, and strange quark matter, defined by unique equations of state and stability criteria.
The use of the TOV equation with density-dependent bag constants, finite strange quark mass, and quark pairing models determines maximum masses in the range of 2.0–3.6 M⊙.
Observational data from high-mass pulsars and gravitational wave events provide critical tests for these models, highlighting the interplay between microphysical parameters and modified gravity effects.

Strange stars are hypothesized compact stellar objects composed entirely of deconfined strange quark matter, where up, down, and strange quarks are in equilibrium under beta decay, potentially stabilized at high densities. The maximum mass limit of such stars is a key factor for distinguishing them from neutron stars and black holes, and serves as a critical test of quantum chromodynamics (QCD) and possible modifications of gravity in the strong-field regime. The determination of this limit is governed by the equation of state (EOS), microphysical parameters such as strange quark mass, the potential modification of the bag constant with density, the inclusion of higher-order interactions, rotation, magnetic field effects, and the underlying gravitational framework (either General Relativity or various modifications thereof). Recent theoretical advances—including models involving density-dependent bag functions, higher-order curvature-matter couplings, and new EOS based on color superconductivity—point to a diverse theoretical landscape for the astrophysically relevant mass ceiling of strange stars.

1. Governing Equations and Equations of State

The primary theoretical apparatus for establishing the mass limit of strange stars is the Tolman–Oppenheimer–Volkoff (TOV) equation integrated with a suitable EOS. The MIT bag model is the standard baseline: $p = \frac{1}{3}(\rho - 4B)$ where $p$ is pressure, $\rho$ is energy density, and $B$ is the bag constant reflecting QCD vacuum energy. Extensions of the MIT bag model commonly introduce (i) a density-dependent bag constant $B(n)$ —motivated by the expectation that the distinction between perturbative and nonperturbative vacua fades at high density, (ii) explicit inclusion of finite strange quark mass $m_s$ , as well as (iii) additional interaction terms or polytropic forms capturing stiffer EOS at high density (Chattopadhyay et al., 21 Jan 2025, Roy et al., 13 May 2025, Wu et al., 2020). For instance, bag constants may be parametrized via Wood–Saxon or exponential forms: $B(n) = B_0 e^{-(a_1 x^2 + a_2 x)}, \quad x = n / n_0$ where $n$ is baryon density and $n_0$ is nuclear saturation density (Roy et al., 13 May 2025).

The stiffness of the EOS may be further increased by including pairing/correlation effects (as in the Color-Flavor Locked (CFL) phase), or by additional repulsive vector interactions. In the CFL phase, quarks form Cooper pairs, leading to modified energy and pressure formulas incorporating a pairing gap parameter $\Delta$ along with $m_s$ , which significantly increases the maximum mass (Goswami et al., 2023). Polytropic EOSs with adjustable exponent $\gamma$ are also used to explore the required stiffness for large mass limits, with self-bound strange stars requiring $\gamma > 1.4$ to reach masses above $2.3\,M_\odot$ (Wu et al., 2020).

2. Microphysical Parameters: Strange Quark Mass and Density Dependence

The finite strange quark mass $m_s$ introduces a softening effect on the EOS, lowering the maximum mass and radius for a given density profile. For example, increasing $m_s$ from 0 to $100\,\mathrm{MeV}$ at fixed bag parameter and baryon density reduces $M_\mathrm{max}$ from $2.01~M_\odot$ to $1.96~M_\odot$ and the corresponding radius from $10.96~\mathrm{km}$ to $10.69~\mathrm{km}$ (Chattopadhyay et al., 21 Jan 2025). There is a direct correlation: higher $m_s$ demands a higher baryon density $n$ to achieve deconfinement and stability against decay into ordinary nuclear matter, as signaled by the condition $E_B < 930.4\,\mathrm{MeV}$ per baryon.

The density dependence of $B(n)$ further modulates the phase structure: at higher $n$ , $B(n)$ decreases (the pressure from confinement wanes), facilitating the transition from hadronic to deconfined quark matter. The precise shape and parametrization of $B(n)$ affect the critical density for phase transitions and consequently the mass–radius relation for strange stars (Roy et al., 13 May 2025). Stable configurations require both $n$ and $B(n)$ to lie within a "window" where the strange quark matter is absolutely stable.

3. Modified Gravity and Curvature-Matter Coupling

The inclusion of higher-order curvature terms and explicit matter–curvature couplings in the action,

$f(\tilde{R},T) = R + \alpha R^2 + 2\beta T,$

with $R$ the Ricci scalar and $T$ the energy-momentum trace, introduces further modifications to the maximum mass (Bhattacharjee et al., 14 Aug 2025). Here, $\alpha$ stiffens the EOS by introducing quadratic curvature corrections, and $\beta$ modulates the coupling between matter and geometry. Negative $\beta$ weakens gravity's effect, favoring higher maximum masses, while positive $\beta$ (strong coupling) reduces the mass limit.

Under this framework, modified TOV equations acquire additional curvature and coupling terms that alter both the mass gradient and pressure gradient equations. The result is a mass–radius relationship highly sensitive to $\alpha$ and $\beta$ . For suitable combinations (e.g., large $\alpha$ , negative or small $\beta$ ), the maximum mass of strange stars can be pushed well above the canonical $2\,M_\odot$ limit established under General Relativity—up to $3.11\,M_\odot$ for appropriately chosen $B_g$ (Bhattacharjee et al., 14 Aug 2025).

Table 1: Maximum Mass as Function of EOS and Gravity Modifications

EOS/Model	Gravity Framework	$M_\mathrm{max}\ [M_\odot]$	Notes
MIT bag ( $m_s=0$ )	GR	$2.01$–$2.03$	$B_g\approx 57$ – $95~\mathrm{MeV}/\mathrm{fm}^3$
Modified MIT bag ( $m_s$ finite, $B(n)$ )	GR	$1.96$–$2.02$	$m_s=100~\mathrm{MeV}$ , $B(n)$ profile
Linked bag/strangeon (stiff, self-bound)	GR	$\sim 2.5$	Strong condensation, $\Lambda_{1.4}\sim180$ –$340$
MIT bag (anisotropic/magnetic/rotating)	GR	$2.3$–$2.8$	Rapid rotation/magnetic enhancement
CFL EOS (color-superconducting phase)	GR	$\sim 3.6$	$m_s=0$ , decreases with finite $m_s$
MIT bag ( $B_g=95$ MeV/fm $^3$ )	$f(\tilde{R},T)$ , $\alpha\gg1$ , $\beta\approx 0$	$3.11$	High-order gravity, soft bag param value

4. Astrophysical Implications and Observational Connections

A principal implication of new mass limits exceeding $2\,M_\odot$ is the compatibility with high-mass pulsars and compact objects discovered in gravitational wave events. For instance, the secondary component in GW190814 (mass $2.5$– $2.67~M_\odot$ ) exceeds the static mass limit predicted by conventional GR-based strange star models, but can be plausibly modeled as a strange star in frameworks incorporating higher-order curvature corrections (Bhattacharjee et al., 14 Aug 2025). Similarly, the “black widow” pulsar PSR J0952–0607 (mass $\sim 2.35~M_\odot$ ) is compatible with models allowing for EOS-stiffening mechanisms such as the CFL phase or rotational enhancement (Goswami et al., 2023, Kayanikhoo et al., 2023). For strange stars, the deformation, compactness, and radius (typically 10–12 km) are generally more compact relative to neutron stars at comparable mass.

Tidal deformabilities predicted for strangeon stars ( $\Lambda_{1.4}\sim180$ –$340$) fall within GW170817’s upper bound ( $\Lambda_{1.4}<580$ ), reinforcing the viability of stiff self-bound configurations (Miao et al., 2020). The presence of a strong electrical field at the quark surface (controlled by mass scaling and surface mass difference parameters) may support a thin crust, affecting pulse profile observations and magnetospheric properties (Li, 2010).

5. Role of Rotation, Magnetic Field, and Anisotropy

Rotation and magnetic fields further modulate stellar structure. Differential rotation (quantified via a parameter $A$ in the KEH rotation law) can theoretically increase the maximum mass of a strange star by a factor of four over its static value, particularly for modest degrees of differential rotation ( $A\simeq0.15$ ). This is more pronounced than for neutron stars and may play an important role in post-merger remnants and rapidly accreting systems (Szkudlarek et al., 2019).

Strong interior magnetic fields (up to $5\times10^{17}$ G), possibly combined with rapid rotation (up to 1200 Hz), can elevate the $M_\mathrm{max}$ further (e.g., $2.8~M_\odot$ ) and induce significant deformation, with equatorial-to-polar radius ratios up to $\sim1.5$ (Kayanikhoo et al., 2023). Anisotropy—arising either from intrinsic microphysical effects, superstrong magnetic fields, or gravitational decoupling—also enhances the maximum mass, with anisotropic pressure configurations supporting heavier and larger stars relative to isotropic ones (Panahi et al., 2015, Maurya et al., 2022).

6. Stability, Viability Criteria, and Energy Conditions

All physically viable strange star configurations must adhere to a range of stability and regularity criteria:

Energy conditions: NEC, WEC, DEC, and SEC are verified throughout the stellar interior.
Causality: The squared sound speed in both radial and tangential directions ( $v_\mathrm{r}^2$ , $v_\mathrm{t}^2$ ) must satisfy $0\leq v^2\leq1$ .
Dynamical stability: The adiabatic index $\Gamma=\frac{\rho+p}{p} \frac{dp}{d\rho}$ exceeds $4/3$ everywhere, ensuring stability against radial oscillations.
Absence of dynamical/“cracking” instabilities: Checks such as $|v_\mathrm{t}^2 - v_\mathrm{r}^2|<1$ (Herrera's condition) are met (Deb et al., 2018).
Tidal deformability and Love number constraints: Predicted values must be consistent with those inferred from gravitational wave observations.

Regularity and monotonicity (e.g., $d\rho/dr<0$ , $dp/dr<0$ from center to surface) are further verified in all accepted models.

7. Summary Table of Key Parametric Dependencies

Parameter/Effect	Trend for $M_\mathrm{max}$	Typical Realistic Range
Strange quark mass $m_s$	$\uparrow m_s \Rightarrow \downarrow M_{\max}$	0–100 MeV
Bag constant $B_g$	$\uparrow B_g \Rightarrow \downarrow M_{\max}$	57.55–114 MeV/fm $^{3}$
EOS stiffness ( $\gamma,\Delta$ )	$\uparrow$ stiffness $\Rightarrow \uparrow M_{\max}$	$\gamma>1.4$ for $M_\mathrm{max}>2.3\,M_\odot$
Gravity mod. ( $\alpha$ )	$\uparrow \alpha \Rightarrow \uparrow M_{\max}$	Model-dependent
Matter coupling ( $\beta$ )	$\uparrow \beta \Rightarrow \downarrow M_{\max}$	Model-dependent
Differential rotation	Present $\Rightarrow \uparrow\uparrow M_{\max}$	up to $4\times$ static mass
CFL phase/pairing $(\Delta)$	Enhanced $\Rightarrow \uparrow M_{\max}$	$\sim3.6\,M_\odot$ (max)

8. Synthesis and Outlook

Current theoretical exploration establishes a broad possible range for the maximum mass of strange stars. Models within conventional GR and a standard MIT bag EOS, considering finite $m_s$ , yield $M_\mathrm{max}\sim 2.0~M_\odot$ . Including density-dependent bag constants, quark pairing (CFL), or higher-order curvature-matter couplings (large $\alpha$ , small or negative $\beta$ ) enables configurations with maximum masses in the range $2.5$– $3.6~M_\odot$ , compatible with recent LIGO/Virgo discoveries (Goswami et al., 2023, Bhattacharjee et al., 14 Aug 2025). Additional enhancements arise from rotation and strong magnetic fields. These results offer a theoretical framework in which the observed “mass gap” objects, bridging the divide between neutron stars and black holes, may be interpreted as massive strange stars, pending further constraints from tidal deformability, radius measurements, and improved EOS calculations.

A plausible implication is that the detection of compact stars in the $2.5$– $3.1\,M_\odot$ mass range, especially with small radii and low tidal deformability, could provide compelling evidence for the existence of strange stars, or at minimum, for the relevance of nonstandard gravitational dynamics or ultra-stiff equations of state at supranuclear densities.