Variation of the $\mathrm{\Delta }$ baryon mass and hybrid star properties in static and rotating conditions

The possible conditions for hadron-quark phase transition in hybrid star cores are investigated in the present work. For the hadronic matter part the effective chiral model is adopted. Exotic baryonic degrees like hyperons and the $\mathrm{\Delta }$ baryons are also taken into account. As $\mathrm{\Delta }\mathrm{s}$ possess Breit-Wigner mass distribution ($1232\pm 120$ MeV), the hadronic equation of state is obtained by varying the mass of the $\mathrm{\Delta }$ baryons in this range. For the quark phase the MIT Bag model is chosen with repulsive effects of the unpaired quarks. Phase transition is achieved using Gibbs construction and the gross properties of the resultant hybrid star are calculated in both static and rotating conditions and compared with the various constraints on them from different observational and empirical perspectives. The work presents a thorough study of the phase transition properties like the critical density of the appearance of quarks, the density range for the persistence of the mixed phase, and the population of different hadrons and quarks in hybrid star matter. The hybrid star properties, calculated in both static and rotating conditions, are found to be consistent with the bounds on them from different perspectives.

Figure 1

Equation of state ($\varepsilon$ vs $P$) of hybrid star matter for different masses of $\mathrm{\Delta }$ baryons. The red (medium gray), green (light gray), and blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The hadron-quark mixed phase regions are also marked by vertical dashed lines.

Figure 2

Relative particle fraction of different baryons and quarks in hybrid star matter for $m_{\mathrm{\Delta }}=1112$ MeV. The shaded region indicates the mixed phase.

Figure 3

Same as Fig. 2 but for $m_{\mathrm{\Delta }}=1232$ MeV.

Figure 4

Same as Fig. 2 but for $m_{\mathrm{\Delta }}=1352$ MeV.

Figure 5

Speed of sound in hybrid star matter for different masses of $\mathrm{\Delta }$ baryons. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The mixed phase regions are also marked with vertical dashed lines. The shaded region shows the bound on speed of sound from GW170817 analysis [49].

Figure 6

Mass-radius relationship for static stars with $\beta$ stable matter [denoted by ‘N’ (thick solid magenta curve)], in the presence of hyperons and $\mathrm{\Delta }\mathrm{s}$ (dotted curves) and hybrid star matter (thin solid curves) for different masses of $\mathrm{\Delta }$ baryons. Among the dotted and thin solid curves the lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. Observational limits imposed from high mass pulsars like PSR $\mathrm{J0348}+0432$ ($M=2.01\pm 0.04M_{\odot }$) [43] (area under horizontal solid cyan lines) and PSR $\mathrm{J0740}+6620$ [$2.{14}_{-0.09}^{+0.10}{M}_{\odot }$ (68.3%—area under horizontal dot-dashed orange lines) and $2.{14}_{-0.18}^{+0.20}{M}_{\odot }$ (95.4%—area under horizontal dashed-double dotted dark green lines)] [44] are also indicated. The horizontal black and brown dashed horizontal lines indicate the canonical mass $M=1.4M_{\odot }$ (GW170817 M1) and mass $M=1.6M_{\odot }$ (GW170817 M2), respectively. The limits on $R_{1.4}$ [45, 46] and $R_{1.6}$ [133] prescribed from GW170817 are indicated by crossmarks. The constraints on the M-R plane from the NICER experiment for PSR $\mathrm{J0030}+0451$ [54] are also compared (95%—outer grey shaded region and 68%—inner pink shaded region).

Figure 7

Baryonic mass vs gravitational mass of hybrid stars for different masses of $\mathrm{\Delta }$ baryons. The rightmost red (medium gray), middle green (light gray), and the leftmost blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The magenta box represent the constraint of [134] on baryonic mass $[M_B=(1.366-1.375)M_{\odot }]$ for pulsar B of binary system PSR J0737-3039 with gravitational mass $[M_G=(1.249\pm 0.001)M_{\odot }]$ [135].

Figure 8

Surface gravitational redshift vs mass of hybrid stars for different masses of $\mathrm{\Delta }$ baryons. The leftmost red (medium gray), middle green (light gray), and the rightmost blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. Observational limits imposed from EXO 07482-676 ($Z_S=0.35$) [136], 1E 1207.4-5209 $[Z_S=(0.12-0.23)]$ [55], and RX J0720.4-3125 ($Z_S=0.{205}_{-0.003}^{+0.006}$) [56] are also indicated.

Figure 9

Mass-radius relationship of hybrid stars for different masses of $\mathrm{\Delta }$ baryons rotating with different rotational frequencies. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively.

Figure 10

Rotational frequency versus gravitational mass for hybrid star for different masses of $\mathrm{\Delta }$ baryons rotating at Kepler velocity. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The frequencies from fast rotating pulsars such as PSR $\mathrm{B1937}+21$ ($\nu =633$ Hz) [137] and PSR J1748-2446ad ($\nu =716$ Hz) [57] and XTE J1739-285 ($\nu =1122$ Hz) [138] are also indicated. Range of gravitational mass ($M=1.{44}_{-0.14}^{+0.15}{M}_{\odot }$) for PSR $\mathrm{J0030}+0451$ with $\nu =205.53$ Hz [54] is shown with the brown dotted line and crossmarks.

Figure 11

Moment of inertia ($I$) as a function of gravitational mass ($M$) of hybrid stars for different masses of $\mathrm{\Delta }$ baryons rotating with frequency $\nu =300$ Hz. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. Constraint from PSR J0737-3039A ($I=1.{15}_{-0.24}^{+0.38}\times {10}^{45}$ g ${\mathrm{cm}}^2$ for $M=1.338M_{\odot }$) [139] is also shown.

Figure 12

Same as Fig. 11 but for $\nu =600$ Hz.

Figure 13

Normalized moment of inertia ($I/M{R}^2$) versus compactness factor ($C=M/R$) of hybrid stars for different masses of $\mathrm{\Delta }$ baryons rotating with frequency $\nu =300$ Hz. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The fitted function of normalized $I$ from various theoretical models for slow rotation (black dashed line) [140] is shown along with the uncertainty region (shaded region) [59].

Figure 14

Same as Fig. 13 but for $\nu =600$ Hz.

Figure 15

Normalized moment of inertia ($I/M^3$) versus compactness factor ($C=M/R$) of hybrid stars for different masses of $\mathrm{\Delta }$ baryons rotating with frequency $\nu =300$ Hz. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The lower red (medium gray), middle green (light gray), and the upper blue (deep gray) curves correspond to $m_{\mathrm{\Delta }}=1112$, 1232, and 1352 MeV, respectively. The fitted value of normalized $I$ from various theoretical models for slow rotation (black dashed line) [140] is shown along with the uncertainty region (shaded region) [59].

Figure 16

Same as Fig. 15 but for $\nu =600$ Hz.

Figure 17

Variation of maximum gravitational mass with Bag constant.

Figure 18

Variation of radius with Bag constant.