Cold Strange Quark Star

• Cold strange quark star is a self-bound compact object composed predominantly of up, down, and strange quarks in weak-interaction equilibrium at temperatures well below 1 MeV.
• Its stellar structure and mass–radius relations are derived from the TOV equations using the MIT bag model, yielding maximum masses around 1.7 M⊙ and radii near 9 km.
• Observational diagnostics include high surface redshifts, absence of atomic lines due to a bare quark surface, and unique gravitational-wave signatures distinguishing it from neutron stars.

A cold strange quark star is a self-bound compact stellar object composed predominantly of up, down, and strange quarks in weak-interaction and chemical equilibrium, existing at temperatures negligible compared to the quark Fermi energy, typically $T \ll 1$ MeV. These objects are supported against gravitational collapse by quark degeneracy pressure rather than nuclear forces, and their global and microphysical properties are dictated by the underlying equation of state (EoS) of strange quark matter (SQM), as described by quantum chromodynamics (QCD)-motivated models. The cold limit is relevant for mature compact stars whose internal temperatures are far below the Fermi temperature and for which all thermal corrections to the EoS are negligible on nuclear energy scales (Goncalves et al., 2020).

1. Equation of State and Thermodynamic Stability

In the standard framework, the MIT bag model provides the baseline EoS for SQM at $T=0$. For a gas of massless $u$, $d$, and $s$ quarks, the pressure, energy density, and baryon density are given by

$$P = -B + \frac{1}{4\pi^2}\sum_{f=u,d,s} \mu_f^4,\qquad \epsilon = B + \frac{3}{4\pi^2}\sum_{f=u,d,s} \mu_f^4,$$

where $B$ is the bag constant simulating nonperturbative QCD confinement. The linear relation

$P(\epsilon) = \frac{1}{3} (\epsilon - 4B)$

holds for the strictly massless case. Retaining finite quark masses, e.g., $m_s \sim 100$--150 MeV, modifies the EoS with logarithmic and algebraic mass-dependent terms but preserves the fundamental self-bound property (Goncalves et al., 2020).

Absolute stability of SQM is governed by the Bodmer–Witten–Terazawa criterion, requiring the energy per baryon $E/A$ in charge-neutral, $\beta$-equilibrated quark matter at zero pressure to satisfy $(E/A)_{\mathrm{SQM}} < 930\,\mathrm{MeV}$, the binding energy per nucleon of ${}^{56}$Fe. For typical $B^{1/4}$ in the range 145–163 MeV, this criterion is fulfilled, and SQM is more stable than iron, implying that cold strange quark stars could exist as the true ground state for compact objects (Goncalves et al., 2020).

2. Stellar Structure and Mass–Radius Relations

The global structure of a cold strange quark star is determined by solving the Tolman–Oppenheimer–Volkoff (TOV) equations with the chosen EoS,

$$\frac{dm}{dr} = 4\pi r^2 \epsilon(r),\qquad \frac{dp}{dr} = -\frac{G\,\epsilon(r)\,m(r)}{r^2}\left[1+\frac{p(r)}{\epsilon(r)}\right] \left[1+\frac{4\pi r^3 p(r)}{m(r)}\right]\left[1-\frac{2G m(r)}{r}\right]^{-1},$$

subject to $m(0)=0$, $p(0)=p_c$ (central pressure), and $p(R)=0$ (defines radius $R$ and mass $M=m(R)$). Stability of equilibrium requires $dM/dp_c>0$ (Goncalves et al., 2020).

Representative results for mass and radius, taking $B^{1/4}=155$ MeV and $m_s=100$ MeV, yield

• Maximum mass $M_{\max} \simeq 1.695\,M_\odot$ at $R\simeq 9.31$ km (finite $m_s$).
• The mass–central-pressure curve turns over beyond $p_c \simeq 396\,\mathrm{MeV}\,\mathrm{fm}^{-3}$, and subsolar-mass configurations ($M<1 M_\odot$) are self-bound primarily by the bag pressure (Goncalves et al., 2020).

Alternative EoS formulations incorporating QCD perturbative corrections generally soften the EoS for nonzero strange quark mass and strong coupling, but, provided the Bag constant remains within the stability window, maximum masses comparable to or exceeding those found in the MIT bag model are preserved (Restrepo et al., 24 Jan 2025, Yang et al., 2023, 0912.1856).

3. Surface Properties, Crusts, and Electrosphere

Cold strange quark stars are sharply self-bound: the quark matter pressure vanishes at a finite baryon number density, producing a uniform-density interior with a well-defined surface. The surface is typically covered only by a thin electrosphere, yielding ultra-high surface electric fields in the range $10^{18}$–$10^{19}$ V/cm. Crusted configurations are possible, featuring a nuclear crust supported above the quark core by the electric field and separated by a gap of a few hundred femtometers. The maximal crust mass does not exceed $10^{-5}M_\odot$ and contributes a radial extension of $0.1$–$1$ km, more pronounced for lower-mass stars (Weber et al., 2012, Xia et al., 3 Nov 2025).

In color-flavor-locked (CFL) quark matter, the electrosphere may vanish due to near-cancellation of bulk charge, but for unpaired or less symmetric configurations, the electron layer persists and plays a crucial role in surface emission and crust physics (Weber et al., 2012).

4. Magnetization, Spin Polarization, and Anisotropy

Inclusion of strong magnetic fields modifies the EoS and star structure. For fields $B \lesssim 10^{18}$–$10^{19}$ G, the energy density includes a magnetization term, and pressure becomes anisotropic for $B\gtrsim2\times10^{18}$ G. Spin polarization, quantified by polarization parameter $\xi$, stiffens the EoS, but overall increases binding due to the magnetic contribution $-\rho\mu\xi B$, leading to marginally decreased $M_{\max}$ and $R_{\max}$, and enhanced stability. Explicit results for $B=5\times10^{18}$ G:

• $M_{\max}$ decreases from $1.35\,M_\odot$ (unmagnetized) to $1.31\,M_\odot$;
• Corresponding radii shrink from $7.60$ km to $7.53$ km (Bordbar et al., 2011);
• For $B=10^{18}$ G, maximum masses in perturbative QCD EoS exceed $3\,M_\odot$ with radii $\sim16.5$–17 km, demonstrating that self-bound, magnetized strange quark stars can be extremely massive (Sedaghat et al., 2022).

Magnetized star models confirm that pure quark star configurations with strong fields remain horizonless (redshift $z<1$) and satisfy the Buchdahl–Bondi bound, never reaching the black hole limit (Sedaghat et al., 2022).

5. Evolution, Compositional Structure, and Cooling

After formation, a proto-strange quark star cools rapidly by neutrino emission (via direct Urca-like processes $d\to u+e^-+\bar{\nu}_e$), with $\nu$-emissivities of order $$10^{26-28} (T/10^9\,\mathrm{K})^6\,\mathrm{erg}\,\mathrm{cm}^{-3}\,\mathrm{s}^{-1}$$. Color-superconducting gaps suppress these channels below $T_c\sim\Delta/1.76$. As the star becomes transparent to neutrinos and cools to $T\ll1\ \mathrm{MeV}$, only photons radiate from the electrosphere or crust. Particle fractions in chemical equilibrium converge to $Y_{u}\approx Y_{d}\approx Y_{s}\approx1/3$ at high density, with an electron fraction vanishing in the cold limit (Chen et al., 16 Jan 2026).

Evolution from proto-SQS to cold SQS involves mass loss via neutrino emission, shrinkage of radius, and an increase in central density. For fixed baryonic mass $M_B=2.40\,M_\odot$, the gravitational mass falls by $\sim0.14\,M_\odot$ and the radius decreases by $\sim0.9$ km as the star cools and deleptonizes (Chen et al., 16 Jan 2026).

6. Observational Diagnostics and Constraints

Cold strange quark star models produce mass–radius relations compatible with $2\,M_\odot$ pulsar measurements for appropriate EoSs (e.g., bag constants $B^{1/4}\approx150$–155 MeV), and can also explain compact objects with unusually low mass and small radius, as inferred for HESS J1731–347 ($M\sim0.8$–1.1$\,M_\odot$, $R\sim10$–11 km), which standard neutron star models cannot accommodate (Clemente et al., 2022).

Other distinguishing features include:

• High surface redshifts ($z\sim 0.2$–$0.3$ for $M=1.4\,M_\odot$, $R=11$ km),
• Absence of hydrogen/helium atomic lines in X-ray spectra (bare quark surface),
• Absence or suppression of glitches and starquakes mediated by a solid quark crust,
• Tidal deformability and merger signatures consistent with current gravitational-wave constraints (Xia et al., 3 Nov 2025, Chen et al., 16 Jan 2026).

In summary, cold strange quark stars, described within the $T=0$ MIT bag model or its QCD-improved generalizations, form a theoretically consistent class of self-bound relativistic objects whose mass–radius relations, cooling properties, and surface phenomena can account for both the most massive and most compact stellar remnants observed to date, potentially providing a distinct signature of absolutely stable strange quark matter in Nature (Goncalves et al., 2020, Chen et al., 16 Jan 2026, Xia et al., 3 Nov 2025).