Robustness of third family solutions for hybrid stars against mixed phase effects

We investigate the robustness of third family solutions for hybrid compact stars with a quark matter core that correspond to the occurrence of high-mass twin stars against a softening of the phase transition by means of a construction that mimics the effects of pasta structures in the mixed phase. We consider a class of hybrid equations of state that exploits a relativistic mean-field model for the hadronic as well as for the quark matter phase. We present parametrizations that correspond to branches of high-mass twin star pairs with maximum masses between $2.05M_{\odot }$ and $1.48M_{\odot }$ having radius differences between 3.2 and 1.5 km, respectively. When compared to a Maxwell construction with a fixed value of critical pressure $P_c$, the effect of the mixed phase construction consists in the occurrence of a region of pressures around $P_c$ belonging to the coexistence of hadronic and quark matter phases between the onset pressure at $P_H$ and the end of the transition at $P_Q$. The maximum broadening which would still allow mass-twin compact stars is found to be ${(P_Q-P_H)}_{\max }\approx P_c$ for all parametrizations within the present class of models. At least the heavier of the neutron stars of the binary merger GW170817 could have been a member of the third family of hybrid stars. We present the example of another class of hybrid star equations of state for which the appearance of the third family branch is not as robust against mixed phase effects as that of the present work.

Figure 1

The mixed phase construction (solid line) for given hadronic matter (dashed line) and quark matter (dash-dotted line) EoS models employed in this work. The Maxwell construction corresponds to switching at ${\mu }_c$ from the hadronic EoS for ${\mu }_B<{\mu }_c$ to the quark matter EoS for ${\mu }_B>{\mu }_c$.

Figure 2

Hybrid EoS according to the mixed phase construction of Fig. 1 described in the text, for the string screening parameter values $\alpha =0.18$ and 0.22. For the mixed phase parameter ${\mathrm{\Delta }}_P=\mathrm{\Delta }P/P_C$ the values 0.0 (Maxwell construction limit) 0.03, 0.05, and 0.07 are chosen.

Figure 3

Same as Fig. 2 for pressure vs energy density.

Figure 4

Speed of sound squared as a function of energy density, derived from the set of EoSs in Fig. 3 for $\alpha =0.22$ and different mixed phase parameters ${\mathrm{\Delta }}_P=0.0\text{–}0.07$.

Figure 5

Mass-radius diagram for the hybrid star sequences of this work for different values of the screening parameter, $\alpha =0.18$, 0.20, 0.22, 0.24, and varying value of the mixed phase parameter: ${\mathrm{\Delta }}_P=0$ (solid lines; the disconnected NS and third family branches are joined by a thin dotted line representing unstable configurations to guide the eye), 0.03 (dash-double-dotted lines), 0.05 (dotted lines), and 0.07 (double-dash-dotted lines). The critical value for the transition from the third family case to a connected hybrid star branch is ${\mathrm{\Delta }}_P\sim 0.06$ in all cases.

Figure 6

Mass-radius diagram for the hybrid star sequences of this work for the screening parameter $\alpha =0.3$ and varying value of the mixed phase parameter: ${\mathrm{\Delta }}_P=0$ (solid line), 0.03 (dash-double-dotted line), 0.05 (dotted line), 0.07 (double-dash-dotted line), and 0.10 (long-dashed line). The critical value for the transition from the third family case to a connected hybrid star branch is ${\mathrm{\Delta }}_P\sim 0.06$. The mass ranges ($m_1$ and $m_2$) for the two neutron stars in the merger event GW170817 [5] indicate that the heavier could be a third family member while the lighter one is a normal neutron star.

Figure 7

Hybrid star EoS of Benic et al. [11] with the mixed phase construction of the present work applied to it for different values of the mixed phase parameter: ${\mathrm{\Delta }}_P=0$ (bold solid line), 0.01 (thin solid line), 0.02 (short-dashed line), 0.03 (double-dash-dotted line), 0.05 (dash-dotted line), and 0.07 (long-dashed line).

Figure 8

Mass-radius diagram for the hybrid star branches obtained with the EoS shown in Fig. 7 for different values of the mixed phase parameter ${\mathrm{\Delta }}_P$; line styles are the same as in that figure. The critical value for the transition from the third family case to a connected hybrid star branch is ${\mathrm{\Delta }}_P\sim 0.02$.