Cooling of Compact Stars with Color Superconducting Phase in Quark Hadron Mixed Phase

We present a new scenario for the cooling of compact stars considering the central source of Cassiopeia A (Cas A). The Cas A observation shows that the central source is a compact star that has high effective temperature, and it is consistent with the cooling without exotic phases. The observation also gives the mass range of $M \geqslant 1.5 M_\odot$, which may conflict with the current plausible cooling scenario of compact stars. There are some cooled compact stars such as Vela or 3C58, which can be barely explained by the minimal cooling scenario, which includes the neutrino emission by nucleon superfluidity (PBF). Therefore, we invoke the exotic cooling processes, where a heavier star cools faster than lighter one. However, the scenario seems to be inconsistent with the observation of Cas A. Therefore, we present a new cooling scenario to explain the observation of Cas A by constructing models that include a quark color superconducting (CSC) phase with a large energy gap; this phase appears at ultrahigh density region and reduces neutrino emissivity. In our model, a compact star has CSC quark core with a low neutrino emissivity surrounded by high emissivity region made by normal quarks. We present cooling curves obtained from the evolutionary calculations of compact stars: while heavier stars cool slowly, and lighter ones indicate the opposite tendency without considering nucleon superfluidity. Furthermore, we show that our scenario is consistent with the recent observations of the effective temperature of Cas A during the last 10 years, including nucleon superfluidity.

💡 Research Summary

The paper addresses a long‑standing tension in neutron‑star cooling theory: the young compact object in the Cassiopeia A (Cas A) supernova remnant exhibits a relatively high effective surface temperature (T_eff) for its age (~330 yr) and a mass inferred to be ≳ 1.5 M_⊙. Standard “minimal cooling” models, which include only modified Urca, bremsstrahlung, and nucleon superfluidity (pair‑breaking‑formation, PBF), can reproduce the rapid temperature decline observed over the past decade only if a neutron‑triplet (3P₂) superfluid transition occurs. However, these models also predict that more massive stars, having higher central densities, should cool faster because exotic neutrino‑emitting processes (e.g., direct Urca, hyperon Urca) become operative. This prediction conflicts with the high temperature of the massive Cas A object, while other relatively cool objects (Vela, 3C 58, SAX J1808) appear to require stronger cooling than minimal models provide.

To reconcile these disparate observations, the authors construct a hybrid star model that contains a quark–hadron mixed phase (MP) in the core and explicitly incorporates a color‑superconducting (CSC) quark phase. The mixed phase is built using a Wigner‑Seitz cell approximation with an MIT bag description for quark matter (bag constant B = 100 MeV fm⁻³, strong coupling α_s = 0.2) and a Brueckner‑Hartree‑Fock (BHF) equation of state for hadronic matter (hyperons are omitted for simplicity). Surface tension is taken as σ = 40 MeV fm⁻². The resulting EoS yields a maximum stellar mass of 1.53 M_⊙ and radii ≈ 8.6 km, compatible with the lower bound of Cas A’s inferred mass but below the recent 2 M_⊙ pulsar measurements; the authors note that more sophisticated EoS could raise the maximum mass.

In the MP, the volume fraction F of quark matter varies with radius; the authors adopt a simple prescription that the neutrino emissivity from quarks is ε_ν = F ε_ν,0, where ε_ν,0 is the standard Iwamoto (1980) quark direct‑Urca rate. Crucially, they introduce a critical volume fraction F_C (0.10–0.20). Where F > F_C, the quark component is assumed to enter a CSC state with a large pairing gap Δ ≫ k_B T (Δ ≈ 10 MeV). In this regime, quark β‑decay neutrino emission is exponentially suppressed, ε_ν ∝ exp(−Δ/k_B T), effectively turning off quark cooling in the innermost core. The surrounding layer, where F < F_C, remains in a normal quark phase and continues to emit neutrinos via direct Urca. Because the thickness of this normal‑quark shell shrinks as the central density (and thus stellar mass) increases, more massive stars possess a larger CSC core and a thinner neutrino‑emitting shell, leading to slower overall cooling. Conversely, lighter stars have a more extended normal‑quark region and cool more rapidly.

The authors compute cooling curves for three representative masses (1.03, 1.32, 1.50 M_⊙) with corresponding central densities 1.48, 1.82, and 2.67 × 10¹⁵ g cm⁻³. By varying F_C they obtain three families of curves. For F_C ≈ 0.125, the 1.50 M_⊙ model cools more slowly than the 1.03 M_⊙ model, reproducing the observed high T_eff of Cas A while allowing the lighter models to match the cooler temperatures of objects like 3C 58. However, the Vela data (which provide a lower bound on T_eff) are not satisfied by the default quark emissivity. To address this, the authors artificially reduce the normal‑quark emissivity by factors of 0.1 and 0.01, arguing that physical mechanisms such as an increased strange‑quark fraction, reduced electron density in the MP, or the presence of a 2SC pairing pattern could naturally lower the neutrino rate. With a 0.01 reduction, the Vela temperature is accommodated.

In addition to quark physics, the authors incorporate nucleon superfluidity. They adopt a phenomenological critical temperature profile for neutron 3P₂ pairing, tuned to reproduce the observed ~2–3 % decline in Cas A’s surface temperature over the past decade (Heinke & Ho 2010). Only the neutron triplet superfluidity is considered; singlet pairing of neutrons and protons is neglected because it operates at lower densities or is less certain. The surface composition is assumed to be carbon with a thin helium layer, which raises the early‑time T_eff and improves agreement with the data.

The discussion acknowledges several uncertainties: (i) the chosen EoS does not reach the 2 M_⊙ constraint, (ii) the exact density threshold for CSC, the magnitude of Δ, and the detailed composition of the MP (u, d, s quark fractions, electron chemical potential) remain poorly known, (iii) the treatment of the mixed‑phase geometry (droplets, rods, slabs, tubes, bubbles) is simplified, and (iv) the impact of possible pseudo‑Nambu‑Goldstone bosons in the CFL phase on neutrino emission is ignored. Nevertheless, the authors argue that the essential qualitative feature—mass‑dependent suppression of quark neutrino emission by a large CSC gap—offers a natural explanation for the apparently contradictory cooling behavior of massive hot stars and lighter cool stars.

Finally, the paper points to future experimental avenues. Intermediate‑energy heavy‑ion facilities such as J‑PARC and GSI could probe the equation of state at high baryon density, potentially constraining hyperon appearance, meson condensation, and the onset of deconfined quark matter and CSC phases. Combined with improved X‑ray observations of young neutron stars, these efforts could refine the parameters (F_C, Δ, bag constant, surface tension) needed for quantitative cooling models.

In summary, by embedding a color‑superconducting quark core within a quark–hadron mixed phase, and by allowing the CSC region to quench neutrino emission, the authors present a unified cooling scenario that simultaneously accounts for the high temperature of the massive Cas A neutron star, the rapid temperature decline over the last decade (via nucleon 3P₂ superfluidity), and the lower temperatures of lighter neutron stars such as Vela and 3C 58. This work highlights the importance of exotic phases of dense matter in neutron‑star thermal evolution and motivates further theoretical and observational studies of color superconductivity in compact objects.