Rotational enhancement and stability of protoquark stars during thermal evolution

TL;DR

The paper addresses how rotation and thermal evolution affect protoquark stars and their observable signatures. It develops an isentropic, density-dependent quark-mass EOS within the DDQM framework and computes rotating equilibrium sequences using the rns code across four evolutionary stages from hot, lepton-rich to cold, catalyzed matter, extracting global properties and energy partitions. The main findings show that uniform rotation can boost the maximum mass by up to $\sim 40\%$, increase the equatorial radius $R_e$, angular momentum $J$, moment of inertia $I$, and quadrupole moment $|Q|$, with $T_{\rm kin}/|W|$ reaching $\sim 0.18$--$0.19$ near the Kepler limit, indicating susceptibility to gravitational-wave–emitting instabilities; thermal evolution contracts the star and reduces these quantities as deleptonization proceeds. These results reveal distinct rotational and thermal histories for protoquark stars compared to hadronic stars and imply that combined mass–radius–spin–gravitational-wave data from multimessenger observations are essential to infer the interior composition and confirm the presence of quark matter in compact objects.

Abstract

We present the first systematic study of rigidly rotating protoquark stars based on isentropic equations of state (EOS) within the density-dependent quark mass (DDQM) framework. Using a quasi-static equilibrium approach, we follow the Kelvin--Helmholtz evolution from hot, lepton-rich matter to a cold, catalyzed quark star. Rotation substantially enhances the maximum stable mass (by up to $\sim 40\%$), equatorial radius, and key rotational observables, with the ratio of rotational kinetic to gravitational potential energy, $T_{\rm kin}/|W|$, reaching $0.18$--$0.19$ near the Keplerian limit, indicating a heightened susceptibility to gravitational-wave--emitting instabilities. Thermal evolution introduces a clear ordering: all stellar properties peak during the lepton-rich stages and decrease monotonically as the star cools. Compared to hadronic stars, rotating protoquark stars exhibit larger radii, higher moments of inertia, and stronger quadrupolar deformation, producing a distinct signature in the mass--radius--spin plane that can accommodate objects such as HESS~J1731--347 and PSR~J0740+6620. These results demonstrate that future multimessenger observations must account for both thermal history and rotation to robustly identify quark matter in compact stars.

Figures (8)

• Figure 1: The gravitational mass $M$ of rotating protoquark stars as a function of equatorial radius $R_e$. In all four panels, the small circles along each curve represent the structural evolution of a single star with a fixed baryonic mass of $M_b = 1.55\,M_\odot$. This serves as the baseline configuration, corresponding to $\sim 1.45\, M_\odot$ cold, catalyzed neutron star. In other cases, this gravitational mass can differ depending on the conditions and can be clearly seen in the figure. In the $T=0$ panel, we also have observational constraints, i.e the steel blue area indicates the constraints obtained from the binary components of GW170817, with their respective 90% and 50% credible intervals. Additionally, the plot includes the 1 $\sigma$ (68%) CI for the 2D mass-radius posterior distributions of the millisecond pulsars PSR J0030 + 0451 (in cyan and yellow color) riley2019Miller:2019cac and PSR J0740 + 6620 (in orange and peru color)riley2021Miller:2021qha, based on NICER X-ray observations. Furthermore, we display the latest NICER measurements for the mass and radius of PSR J0437-4715 Choudhury:2024xbk (lilac color). The supernova remnant HESS J1731$-$347 2022NatAs...6.1444D is shown in red, with the outer contour representing the 90% CL and the inner contour representing the 50% CL.
• Figure 2: The core temperature evolution of rotating proto stars examined along gravitational-mass sequences. The stars in the plots indicate the maximum mass for each configuration. The small open circles in the curve show the structural evolution of a single star with fixed baryonic mass of 1.55 $\rm M_\odot$.
• Figure 3: Variation of the ratio of the rotational frequency to the Keplerian frequency $\nu/\nu_K$ as a function of the stellar mass.
• Figure 4: The moment of inertia $I$ as a function of the stellar mass for different rotational fattening. Overlaid error bars in panel $T=0$ represent observational constraints from the following pulsars: Red: J0437$-$4715, Blue: J0751+1807, Green: J1713+0747, Orange: J1802$-$2124, Purple: J1807$-$2500B, Brown: J1909$-$3744, Pink: J2222$-$0137, Gray: J0740+6620 (NICER), Li_2022 Olive: J0030+0451 (NICER), Silva:2020acr Cyan: J0737$-$3039A (Double Pulsar) Kumar:2019xgpPhysRevD.105.063023Bejger:2005jy. The $I$ increases along the stable mass sequence up to $M_{\rm max}$ (starred points along the curve) and then decreases as the mass decreases along the unstable branch.
• Figure 5: Variation of the angular momentum, $J$, as a function of the stellar mass for different rotational flattening.