Study of anisotropic compact stars in $f({\mathcal {R}},{\mathcal {T}},{\mathcal {R}}_{\chi \xi }{\mathcal {T}}^{\chi \xi })$ gravity

Abstract

This paper aims to examine the composition of various spherically symmetric star models which are coupled with anisotropic configuration in $f({\mathcal {R}},{\mathcal {T}},{\mathcal {Q}})$ gravity, where ${\mathcal {Q}}={\mathcal {R}}_{\chi \xi }{\mathcal {T}}^{\chi \xi }$. We discuss the physical features of compact objects by employing bag model equation of state and construct the modified field equations in terms of Krori–Barua ansatz involving the unknowns (A, B, C). The observational data of 4U 1820-30, Vela X-I, SAX J 1808.4-3658, RXJ 1856-37 and Her X-I are used to calculate these unknowns and bag constant $\mathfrak {B}_{{\mathfrak {c}}}$. Further, we observe the behaviour of energy density, radial and tangential pressure as well as anisotropy through graphical interpretation for a viable model ${\mathcal {R}}+\varrho {\mathcal {Q}}$ of this gravity. For a particular value of the coupling constant $\varrho$, we study the behaviour of mass, compactness, red-shift and the energy bounds. The stability of the considered stars is also checked by using two criteria. We conclude that our structure developed in this gravity is in good agreement with all the physical requirements.

Appendix A

The value of adiabatic index in terms of Krori–Barua solution takes the form

$$\begin{aligned} \Gamma&=-\Bigg [3\left\{ Ar^2\left( 4\varrho \mathfrak {B}_{{\mathfrak {c}}} \left( Br^2+2\right) +1\right) \right. \\&\quad \!\!\left. -2\mathfrak {B}_{{\mathfrak {c}}} \left( \varrho +\mathrm {e}^{Ar^2} \left( 8\pi r^2-\varrho \right) \right) +Br^2 \times (1-8\varrho \mathfrak {B}_{{\mathfrak {c}}})\right\} \Bigg ]^{-1}\\&\quad \times [2r^2\left\{ \varrho A^2\mathfrak {B}_{{\mathfrak {c}}} r^2+A\left( \varrho \mathfrak {B}_{{\mathfrak {c}}} \left( 2Br^2+5\right) -2\right) \right. \\&\quad \left. -B\left( 2+\varrho \mathfrak {B}_{{\mathfrak {c}}} \times \left( 1+3Br^2\right) \right) \right\} ]. \end{aligned}$$

The term $|v_{s\bot }^2-v_{sr}^2|$ in modified gravity becomes

$$\begin{aligned}&|v_{s\bot }^2-v_{sr}^2|\\&\quad =\frac{1}{12}\bigg |\bigg [\big (\varrho \big (-Ar^2\big (Br^2+2\big )+B^2r^4+3Br^2+1\big )\\&\qquad +\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\big )^2\\&\qquad \times \big \{\varrho A^3r^4\big (8\varrho \mathfrak {B}_{{\mathfrak {c}}}+2\mathfrak {B}_{{\mathfrak {c}}} \mathrm {e}^{Ar^2}\big (r^2-\varrho \big )+1\big )\\&\qquad +A^2r^2\big (\mathrm {e}^{Ar^2} \big (2\varrho (2-3\varrho \mathfrak {B}_{{\mathfrak {c}}})\\&\qquad +4\varrho \mathfrak {B}_{{\mathfrak {c}}}Br^4 -4r^2\big (-2\varrho \mathfrak {B}_{{\mathfrak {c}}}\\&\qquad +\varrho ^2\mathfrak {B}_{{\mathfrak {c}}}B+1\big )\big ) +\varrho \big (Br^2(11-12\varrho \mathfrak {B}_{{\mathfrak {c}}}) -4\varrho \mathfrak {B}_{{\mathfrak {c}}}\big )\big )\\&\qquad +A\big (\varrho \big (-B^2\big )r^4\big (36\varrho \mathfrak {B}_{{\mathfrak {c}}}+6\mathfrak {B}_{{\mathfrak {c}}} \mathrm {e}^{Ar^2}\big (r^2-\varrho \big )-7\big )\\&\qquad -2Br^2\big (4\varrho ^2\mathfrak {B}_{{\mathfrak {c}}} +\mathrm {e}^{Ar^2}\big (r^2(3\varrho \mathfrak {B}_{{\mathfrak {c}}}+2) -\varrho (5\varrho \mathfrak {B}_{{\mathfrak {c}}}+2)\big )\big )\\&\qquad +(5\varrho \mathfrak {B}_{{\mathfrak {c}}}-2) \big (\mathrm {e}^{Ar^2}-1\big ) 2\varrho \big )\\&\qquad +\varrho B\big (2\varrho \mathfrak {B}_{{\mathfrak {c}}} +2\mathrm {e}^{Ar^2}\big (-\varrho \mathfrak {B}_{{\mathfrak {c}}} +3\mathfrak {B}_{{\mathfrak {c}}}Br^2\big (r^2-2\varrho \big )-2\big )\\&\qquad +3B^2r^4(8\varrho \mathfrak {B}_{{\mathfrak {c}}}-1) +12\varrho \mathfrak {B}_{{\mathfrak {c}}}Br^2+4\big )\big \}\bigg ]^{-1}\\&\qquad \times \bigg [2\big (\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )+\big (B^2r^4\\&\qquad +3Br^2-A\big (Br^2+2\big )r^2+1\big )\varrho \big )\big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\\&\qquad +\big (-A^2r^4+3B^2r^4 \!+\! 17Br^2-A\big (10Br^2+21\big )r^2\!+\! 4\big )\varrho \big )\\&\qquad \times \big \{r^2\varrho \big (3B(22\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^2+88\varrho \mathfrak {B}_{{\mathfrak {c}}}\\&\qquad -4\mathrm {e}^{Ar^2}\big (3r^2+\varrho \big ) \mathfrak {B}_{{\mathfrak {c}}}+14\big )A^3\\&\qquad +\big (\varrho \big (3B^2(10\varrho \mathfrak {B}_{{\mathfrak {c}}}+9) r^4+2Br^2(31 -36\varrho \mathfrak {B}_{{\mathfrak {c}}})\\&\qquad -8\varrho \mathfrak {B}_{{\mathfrak {c}}}+11\big ) -4\mathrm {e}^{Ar^2}\big (Br^4+(-B\varrho +5\mathfrak {B}_{{\mathfrak {c}}}\varrho +1)r^2\\&\qquad +\varrho (2\varrho \mathfrak {B}_{{\mathfrak {c}}}-1)\big )\big ) A^2+\big (4\mathrm {e}^{Ar^2}\big (B^2\big (r^2-\varrho \big ) (\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^2\\&\qquad -B\varrho +\varrho \mathfrak {B}_{{\mathfrak {c}}}-1\big ) -B\varrho \big (39B^2(2\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^4\\&\qquad +2B(64\varrho \mathfrak {B}_{{\mathfrak {c}}}+55)r^2 +8\varrho \mathfrak {B}_{{\mathfrak {c}}}+69\big )\big )A\\&\qquad +B\big (\mathrm {e}^{Ar^2}\big (4B\big (2r^2-\varrho \big ) (\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)+8\big )\\&\qquad +B\varrho \big (9B^2(6\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^4+2B(28\varrho \mathfrak {B}_{{\mathfrak {c}}}+33)r^2\\ \end{aligned}$$

$$\begin{aligned}&\qquad +6(\varrho \mathfrak {B}_{{\mathfrak {c}}}+7)\big )\big )\big \} r^2-2\big (\mathrm {e}^{Ar^2}\big (A\big (r^2-\varrho \big )+1\big )\\&\qquad +\big (B\big (2Br^2+3\big )-2A\big (Br^2+1\big )\big )\varrho \big ) \big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\\&\qquad +\big (-A^2r^4+3B^2r^4+17Br^2-A\\&\qquad \times \big (10Br^2+21\big )r^2+4\big )\varrho \big )\\&\qquad \times \big \{A^3\varrho \big (B(22\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^2+44\varrho \mathfrak {B}_{{\mathfrak {c}}}+7\big )r^4\\&\qquad +A^2\varrho \big (B^2(10\varrho \mathfrak {B}_{{\mathfrak {c}}}+9)r^4\\&\qquad +B(31-36\varrho \mathfrak {B}_{{\mathfrak {c}}})r^2 -8\varrho \mathfrak {B}_{{\mathfrak {c}}}-4\big (3r^2+\varrho \big )\\&\qquad \times \mathfrak {B}_{{\mathfrak {c}}}\mathrm {e}^{Ar^2}+11\big ) r^2+B\big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\\&\qquad \times \big (B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^2+2\big ) +\varrho \big (3B^3r^6\\&\qquad \times (6\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) +B^2(28\varrho \mathfrak {B}_{{\mathfrak {c}}}+33)r^4\\&\qquad +6B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+7)r^2+8\big )\big ) -A\big (4\mathrm {e}^{Ar^2}\\&\qquad \times \big (r^2-\varrho \big )\big (Br^2 -\varrho \mathfrak {B}_{{\mathfrak {c}}}+1\big )\\&\qquad +\varrho \big (13B^3(2\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^6+B^2(64\varrho \mathfrak {B}_{{\mathfrak {c}}}+55)r^4\\&\qquad +B(8\varrho \mathfrak {B}_{{\mathfrak {c}}}+69)r^2 -4\varrho \mathfrak {B}_{{\mathfrak {c}}}+4\big )\big )\big \} r^2-2\big (\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\\&\qquad +\big (B^2r^4 +3Br^2-A\big (Br^2+2\big )r^2+1\big )\varrho \big )\\&\qquad \times \big (4\mathrm {e}^{Ar^2}\big (A\big (r^2-\varrho \big )+1\big ) +\big (-2A^2r^2+B\big (6Br^2+17\big )\\&\qquad -A\big (20Br^2+21\big )\big )\varrho \big ) \big \{A^3\varrho \big (B(22\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^2+7\\&\qquad +44\varrho \mathfrak {B}_{{\mathfrak {c}}}\big )r^4 +A^2\varrho \big (B^2(10\varrho \mathfrak {B}_{{\mathfrak {c}}}+9) r^4\\&\qquad +B(31-36\varrho \mathfrak {B}_{{\mathfrak {c}}})r^2 -8\varrho \mathfrak {B}_{{\mathfrak {c}}}\\&\qquad -4\mathrm {e}^{Ar^2}\big (3r^2+\varrho \big ) \mathfrak {B}_{{\mathfrak {c}}}+11\big )r^2\\&\qquad +B \big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big ) \big (B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^2+2\big )\\&\qquad +\varrho \big (3B^3(6\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^6+B^2(28\varrho \mathfrak {B}_{{\mathfrak {c}}}+33)r^4\\&\qquad +6B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+7)r^2+8\big )\big )\\&\qquad -A\big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big ) \big (Br^2-\varrho \mathfrak {B}_{{\mathfrak {c}}}+1\big )\\&\qquad +\varrho \big (13B^3(2\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^6+B^2\\&\qquad \times (64\varrho \mathfrak {B}_{{\mathfrak {c}}}+55)r^4 +B(8\varrho \mathfrak {B}_{{\mathfrak {c}}}+69)r^2\\&\qquad -4\varrho \mathfrak {B}_{{\mathfrak {c}}}+4\big )\big )\big \}r^2 +2\big (\mathrm {e}^{Ar^2}\big (r^2 -\varrho \big )\\&\qquad +\big (B^2r^4+3Br^2-A\big (Br^2+2\big )r^2+1\big )\varrho \big ) \big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big )\\&\qquad +\big (4 -A^2r^4+3B^2r^4+17Br^2-A \big (10Br^2+21\big )r^2\big )\varrho \big )\\&\qquad \quad \big \{A^3\varrho \big (B(22\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)\\ \end{aligned}$$

$$\begin{aligned}&\qquad \times r^2+44\varrho \mathfrak {B}_{{\mathfrak {c}}}+7\big ) r^4+A^2\varrho \big (B^2(10\varrho \mathfrak {B}_{{\mathfrak {c}}}+9)r^4\\&\qquad +B(31-36\varrho \mathfrak {B}_{{\mathfrak {c}}})r^2 -8\varrho \mathfrak {B}_{{\mathfrak {c}}}\\&\qquad -4\mathrm {e}^{Ar^2}\big (3r^2+\varrho \big ) \mathfrak {B}_{{\mathfrak {c}}}+11\big )r^2\\&\qquad +B\big (4\mathrm {e}^{Ar^2}\big (r^2-\varrho \big ) \big (B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^2+2\big )\\&\qquad +\varrho \big (3B^3 (6\varrho \mathfrak {B}_{{\mathfrak {c}}}+1)r^6 +B^2(28\varrho \mathfrak {B}_{{\mathfrak {c}}}+33) r^4\\&\qquad +6B(\varrho \mathfrak {B}_{{\mathfrak {c}}}+7) r^2+8\big )\big )\\&\qquad -A\big (4\mathrm {e}^{Ar^2} \big (r^2-\varrho \big )\big (Br^2-\varrho \mathfrak {B}_{{\mathfrak {c}}}+1\big )\\&\qquad +\varrho \big (13B^3(2\varrho \mathfrak {B}_{{\mathfrak {c}}}+1) r^6+B^2 (64\varrho \mathfrak {B}_{{\mathfrak {c}}}+55) r^4\\&\qquad +B(8\varrho \mathfrak {B}_{{\mathfrak {c}}}+69)r^2 -4\varrho \mathfrak {B}_{{\mathfrak {c}}}+4\big )\big )\big \}-4\bigg ]\bigg |. \end{aligned}$$