Rotating Protoquark Stars: Structure & Signatures

• Rotating protoquark stars are compact astrophysical objects formed post-supernova, characterized by rapid rotation and deconfined quark matter under finite-temperature, lepton-rich conditions.
• The study applies CMF and DDQM models with Gibbs construction and relativistic equilibrium methods to accurately determine mass, radius, and stability properties.
• Rotational effects enhance maximum mass, induce significant quadrupole deformation, and elevate gravitational-wave emission, offering unique observational discriminants from hadronic stars.

A rotating protoquark star is a compact astrophysical object formed during the immediate post-bounce phase of a core-collapse supernova, characterized by rapid rotation, intense thermal gradients, and a composition dominated by deconfined up, down, and strange quark matter, often under significant lepton-rich (neutrino-trapped) conditions. These objects either represent genuine quark stars (self-bound) or hybrid stars (where a quark phase coexists with hyperonic or nucleonic matter), with rotational and thermal evolution fundamentally shaping their structure, stability, and observable properties (Roark et al., 2018, Issifu et al., 20 Jan 2026).

1. Equation of State and Composition

The microphysics of rotating protoquark stars is governed by finite-temperature, lepton-rich equations of state (EOS), which determine all macroscopic observables. Two principal models are prevalent:

• The Chiral Mean Field (CMF) model addresses matter consisting of hadrons, hyperons, and quarks under fixed entropy per baryon $S_B$ and electron-lepton fraction $Y_L$ (Roark et al., 2018). The grand potential per unit volume in each phase $X \in \{\text{Hadrons},\, \text{Quarks}\}$ is

$$\frac{\Omega^{(X)}}{V} = -T \sum_{i\in X}\frac{g_i}{(2\pi)^3} \int d^3k\, \ln\left[1+e^{-(E_i^*(k)-\mu_i^*)/T}\right] + \mathcal{V}_{\text{mesons}}^{(X)} + U(\Phi, T, \mu_B)$$

where degrees of freedom, effective masses, chemical potentials, and the order parameter for deconfinement $\Phi$ are dynamically modeled.

• The Density-Dependent Quark Mass (DDQM) model encodes quark confinement and deconfinement via a quark mass formula,

$$m_i(\rho_B, T) = m_{i0} + \frac{D}{\rho_B^{1/3}\left[1+\frac{8T}{\Lambda} e^{-\Lambda/T}\right]^{-1}} + C \rho_B^{1/3}\left[1+\frac{8T}{\Lambda} e^{-\Lambda/T}\right]$$

with calibrated parameters $(C, D, \Lambda)$, reproducing asymptotic freedom at high densities and producing a one-parameter family of isentropic EOS labeled by $(s_B, Y_l)$ (Issifu et al., 20 Jan 2026).

In both frameworks, beta-equilibrium and charge neutrality are imposed, with lepton number either globally or locally conserved. During the earliest stages ($s_B \sim 1$–2, $Y_l \sim 0.4$), deleptonization and thermal cooling drive the system through a Kelvin–Helmholtz epoch, from a hot neutrino-trapped protoquark star to a cold, catalyzed quark star.

2. Phase Structure and Mixed Phase Construction

The deconfinement transition in rotating protoquark stars often involves a non-congruent coexistence of hadronic and quark matter. This is modeled by a Gibbs construction at finite temperature and fixed lepton fraction, in which local charge neutrality is enforced within each phase, but lepton number is maintained globally (Roark et al., 2018).

Key results:

• The total pressure and temperature are continuous across the transition, i.e., $P^{(H)} = P^{(Q)}$ and $T^{(H)} = T^{(Q)}$.
• The baryon density is a volume-fraction-weighted average of the two phases,

$n_B = (1-\chi)\, n_B^{(H)} + \chi\, n_B^{(Q)}$

where $\chi$ is the quark volume fraction.

• Local charge neutrality in the hadronic and quark sectors is imposed separately, resulting in coupled chemical and number density constraints that affect the threshold for quark matter formation.
• No explicit surface tension or Coulomb corrections are introduced beyond the LCN assumption.

Rotation affects the central density and can delay the formation of a mixed phase to lower spin rates, as centrifugal support lowers the maximum core density at fixed baryon number.

3. Rotational Dynamics and Stability Criteria

Rotational equilibria are computed using two complementary approaches:

• The Hartle–Thorne slow-rotation formalism (expanding the spacetime metric and stellar deformation to second order in angular velocity $\Omega$) permits accurate treatment for moderate rotations, as appropriate for large-radius protoquark stars with relatively low Kepler frequencies (Roark et al., 2018).
• Full axisymmetric relativistic equilibria, implemented via the RNS code, enable robust calculation at arbitrary spin rates up to mass-shedding (Kepler) limits for self-bound stars (Issifu et al., 20 Jan 2026).

Critical parameters and diagnostics include:

• Kepler frequency ($\Omega_K$): the maximum spin before mass shedding at the equator.
• The ratio $T_{\rm kin}/|W|$ (rotational kinetic/gravitational binding energy): a diagnostic for rotational instabilities, particularly nonaxisymmetric (CFS) modes. For DDQM stars, $T_{\rm kin}/|W|$ reaches 0.18–0.19 at the Kepler limit ($r_p/r_e = 0.5$), exceeding typical hadronic stars and nearing the instability threshold for secular gravitational-wave emission (Issifu et al., 20 Jan 2026).
• With increasing spin up to $\Omega_K$, rotation can raise the maximum gravitational mass by 3.5% (CMF/hybrid, PNS) to 40% (DDQM/quark star), and similarly enlarge the equatorial radius, moment of inertia, and quadrupole deformation.

4. Rotational and Thermal Evolution: Structural Effects

Detailed numerical results reveal a clear "thermal ordering" of structural properties throughout the protoquark star's evolution:

Thermal Stage	$M_{\max}^{\rm static}$ ($M_\odot$)	$M_{\max}^{\rm rot}$ ($M_\odot$)	$\Delta M_{\max}$	$R_e^{\rm static}$ (km)	$R_e^{\rm rot}$ (km)
Hot, neutrino-trapped ($s_B=1$)	2.33	3.27	+40.3%	14.71	22.33
Intermediate ($s_B=2$)	2.28	3.20	+40.4%	14.36	21.88
Neutrino-transparent ($s_B=2$)	2.24	3.12	+39.3%	14.18	21.59
Cold, catalyzed matter ($T=0$)	2.20	3.05	+38.6%	14.06	21.36

Rotation produces major structural modifications:

• Maximum mass, equatorial radius, moment of inertia, and quadrupole moment are all largest at early, hot stages ($s_B=1$, $Y_l=0.4$) and decrease monotonically with cooling and deleptonization.
• The pure quark-phase core of a hybrid star shrinks by up to 30% in radius under rotation, with its threshold for formation shifting upward in density by $\sim0.2$–0.3 $n_s$ (Roark et al., 2018).
• The mixed-phase layer may be suppressed entirely at high spin rates: for protoquark stars at their Kepler frequency, no extended mixed hadron–quark region is present.
• For fixed gravitational mass, the core temperature drops with increasing spin, e.g., from $\sim 9.4$ MeV (static) to $\sim 6.2$ MeV (mass shedding) at $M=2\,M_\odot$ (Issifu et al., 20 Jan 2026).

5. Distinct Rotational Signatures and Astrophysical Implications

Rotating protoquark stars present rotational and structural signatures that can distinguish them from both cold quark stars and conventional hadronic stars:

• Enhancement of $M_{\max}$ by up to 40%, compared to only $\sim$20% in hadronic models, permits stable compact objects at higher masses and radii.
• Stronger quadrupolar deformation and increased moment of inertia (by up to 20–40% at equal spin), affecting spindown, pulse profiles, and gravitational wave emission.
• Higher $T_{\rm kin}/|W|$ ratios at a given frequency, especially exceeding 0.14 for low $\nu/\nu_K$, indicate increased susceptibility to CFS (Chandrasekhar–Friedman–Schutz) modes and thus early gravitational-wave emission in the post-bounce phase (Issifu et al., 20 Jan 2026).
• In the $M$–$R$–$\nu$ parameter space, rotating quark stars can simultaneously accommodate high-mass objects like PSR J0740+6620 and low-mass, large-radius remnants such as HESS J1731–347, a distinctive signature not naturally achieved by hadronic models with comparable stiffness.

A plausible implication is that rapid, early post-bounce gravitational-wave emission or X-ray pulse-profile studies with NICER or ATHENA could directly probe these structural characteristics, providing a discriminant between hadronic and quark-matter equations of state if thermal and rotational effects are correctly incorporated.

6. Observational Prospects and Future Directions

The intersection of rotational, compositional, and thermal effects mandates multimessenger strategies to robustly identify quark matter in compact stars:

• Third-generation gravitational-wave detectors are expected to probe GW emission from protoquark stars driven by bar-mode (CFS) instabilities, particularly given the elevated $T_{\rm kin}/|W|$ ratios at modest spin rates in self-bound quark EOS (Issifu et al., 20 Jan 2026).
• X-ray timing and modeling campaigns (e.g., with NICER, ATHENA) should incorporate the combined effects of rotationally induced equatorial bulge and thermal inflation when inferring radii and moments of inertia to few-percent accuracy.
• The observed population's distribution in $M$–$R$–$\nu$ space informs EOS constraints, with rotating protoquark stars providing a unifying framework for otherwise discordant data.
• Targeted asteroseismology (r- and f-mode frequencies) and gravitational-wave-driven spindown will further constrain the presence and properties of quark cores, and may resolve the mixed-phase transition features predicted by finite-temperature phase constructions (Roark et al., 2018).

The synthesis of these approaches, and the detailed match of theory to evolving protoquark star models through the Kelvin–Helmholtz epoch, will be central to the identification (or exclusion) of deconfined quark matter in nature’s most extreme compact objects.