Maximum mass of a hybrid star having a mixed phase region in the light of pulsar PSR J1614-2230

Recent observation of pulsar PSR J1614-2230 with mass about 2 solar masses poses a severe constraint on the equations of state (EOS) of matter describing stars under extreme conditions. Neutron stars (NS) can reach the mass limits set by PSR J1614-2230. But stars having hyperons or quark stars (QS) having boson condensates, with softer EOS can barely reach such limits and are ruled out. QS with pure strange matter also cannot have such high mass unless the effect of strong coupling constant or colour superconductivity are taken into account. In this work I try to calculate the upper mass limit for a hybrid stars (HS) having a quark-hadron mixed phase. The hadronic matter (having hyperons) EOS is described by relativistic mean field theory and for the quark phase I use the simple MIT bag model. I construct the intermediate mixed phase using Glendenning construction. HS with a mixed phase cannot reach the mass limit set by PSR J1614-2230 unless I assume a density dependent bag constant. For such case the mixed phase region is small. The maximum mass of a mixed hybrid star obtained with such mixed phase region is $2.01 M_{\odot}$.

💡 Research Summary

The discovery of the millisecond pulsar PSR J1614‑2230 with a precisely measured mass of about 2 M⊙ has become a benchmark for testing the stiffness of the equation of state (EOS) of ultra‑dense matter. In this paper the author investigates whether a hybrid star (HS) that contains a quark–hadron mixed phase can satisfy this stringent mass constraint. The study proceeds in several clearly defined steps.

First, the hadronic sector is modeled with a relativistic mean‑field (RMF) theory that includes the full baryon octet (nucleons plus hyperons). The chosen parameter set (similar to the widely used GM1/TM1 families) reproduces nuclear saturation properties and hyperon potentials, but the appearance of hyperons at a few times nuclear saturation density (ρ₀≈0.16 fm⁻³) softens the EOS considerably.

Second, the deconfined quark phase is described by the simplest version of the MIT bag model. The thermodynamic potential consists of the kinetic contribution of massless u and d quarks, a massive s quark (mₛ≈150 MeV), and a constant bag pressure B. The baseline value B^{1/4}=180 MeV is adopted, which is typical for phenomenological studies. With this EOS a pure quark star reaches a maximum mass of only ~1.6 M⊙, well below the observed pulsar mass.

Third, the transition between the two phases is constructed using the Glendenning construction. Unlike the Maxwell construction, Glendenning’s method imposes global charge neutrality while allowing each phase to carry a net charge. This yields a continuous pressure–density curve and a finite density interval where hadronic and quark matter coexist in a mixed phase. The volume fraction of each component varies smoothly with the baryon density, producing a “pasta‑like” region that can be several ρ₀ wide.

With the fixed‑B EOS for both phases, the Tolman‑Oppenheimer‑Volkoff (TOV) equations are integrated to obtain mass–radius (M–R) sequences. The resulting hybrid stars reach a maximum mass of ≈1.78 M⊙, insufficient to explain PSR J1614‑2230. The author attributes this shortfall to the large mixed‑phase region, which reduces the overall stiffness of the combined EOS.

To overcome the limitation, a density‑dependent bag constant B(ρ) is introduced:

 B(ρ)=B₀ exp