Anisotropic spheres via embedding approach in $f(R,\phi ,X)$ gravity

Abstract

In this manuscript, we investigate the behavior of stellar structure through embedding approach in $f(R, \phi , X)$ modified theory of gravity, where R denotes the Ricci scalar, $\phi$ represents the scalar potential, and X indicates the kinetic potential. For this purpose, we consider the spherically symmetric space-time with anisotropic fluid. We further choose three different stars, i.e., LMC X-4, Cen X-3, and EXO 1785-248 to demonstrate the behavior of stellar structures. We further compare the Schwarzschild space-time as exterior geometry with spherically symmetric space-time to calculate the values of unknown parameters. In this regard, we investigate the graphical features of stellar spheres such as energy density, pressure components, anisotropic component, equation of state parameters, stability analysis and energy conditions. Furthermore, we investigate some extra conditions such as mass function, compactness factor and surface redshift, respectively. The observed behavior of these attributes aligns with accepted physical properties, and the emerging outcomes fall within the observed range. We find that the embedding class one solution for anisotropic compact stars is viable and stable.

Appendix

$$\begin{aligned} t_1(r)= &\, (1+hr^2).\\ t_2(r)= &\, (t_1(r)^2+ohr^2t_1(r)^{\tau }).\\ t_3(r)= &\, 60-3\tau +2(-20+\tau (19+3\tau ))hr^2+(1+\tau )(-4+\tau +\tau ^2)h^2r^4.\\ t_4(r)= &\, 5+hr^2(55-20\tau +(47+4(-4+\tau )\tau )hr^2+(29-12\tau (1+\tau ))h^2r^4).\\ t_5(r)= &\, 3(1+\tau )r^2+(120+\tau (-312+241\tau ))\gamma -3r^{2+2\alpha }\alpha ^2.\\ t_6(r)= &\, (4+\tau )r^2+12(-10+9\tau )\gamma -2r^{2+2\alpha }\alpha ^2.\\ t_7(r)= &\, 3\tau r^2+216\gamma +2\tau (-70+\tau (-87+71\tau ))\gamma -2r^{2+2\alpha }\alpha ^2.\\ t_8(r)= &\, (-1+2\tau )r^2+2(-24+\tau (56+\tau (-27+\tau (-30+17\tau ))))\gamma -r^{2+2\alpha }\alpha ^2.\\ t_9(r)= &\, (20-3r^{2\alpha }\alpha ^2+2h^2r^216(4+3\tau )r^2+420\gamma -16\tau (3+14\tau )\alpha -9r^{2+2\alpha }\alpha ^2)+4h((14+3\tau )r^2+4(27-46\tau )\alpha -\\= &\, 3r^{2+2\alpha }\alpha ^2)+4h^3r^4((2+9\tau )r^2+4(27+\tau (4(5-16\tau )\tau ))\gamma -3r^{2+2\alpha }\alpha ^2)+h^4r^6(4(-1+3\tau )r^2+4(75+4\tau (-36\\= &\, +\tau (21+2\tau -6\tau ^2)))\gamma -3r^{2+2\alpha }\alpha ^2).\\ t_{10}(r)= &\, (1+(-1+\tau )hr^2).\\ t_{11}(r)= &\, 116+4(-3+56\tau )hr^2+(-84+\tau (16+155\tau ))h^2r^4+(-1+\tau )(-44+\tau (68+39\tau ))h^3r^6.\\ t_{12}(r)= &\, 33-28\tau +(-15+4(20-3\tau )\tau )hr^2+(-5+4\tau (-1+9\tau ))h^2r^4.\\ t_{13}(r)= &\, 8+66\tau +(-12+\tau (16+59\tau ))hr^2+(24+7\tau (6+\tau (-1+4\tau )))h^2r^4+2(-1+\tau )(1+\tau (11+\tau (4+3\tau )))h^3r^6.\\ t_{14}(r)= &\, (4+4(1+2\tau )hr^2+16(-1+4\tau )h\gamma -r^{2\alpha }t_{1}(r)\alpha ^2).\\ t_{15}(r)= &\, r^2t_1(r)^6+o^3h^3r^8t_1(r)^{3\tau }+2t_1(r)^4(11+3hr^2(-18+hr^2))\gamma +o^2h^2r^4t_1(r)^{2\tau }(3(r+hr^3)^2+4(59+hr^2(4\\= &\, -39hr^2))\gamma )+ohr^2t_1(r)^{2+\tau }(3(r+hr^3)^2+h(-231+hr^2(194+45hr^2))\gamma ).\\ t_{16}(r)= &\, (t_1(r)^7+8o^4h^4r^6(t_1(r)^4)^{\tau }\gamma +8ht_1(r)^5(-5+hr^2(8+hr^2))\gamma -oh(1+hr^2)^{3+\tau }(-3(r+hr^3)^2+4(\tau +hr^2(-1\\= &\, +hr^2(-52+17hr^2)))\gamma )+o^3h^3r^4t_1(r)^{3\tau }(26\gamma +r^2t_1(r)+6h(6+7hr^2)\gamma ))+o^3h^2r^2t_1(r)^{2+2\tau }(226\gamma +r^2(3-\\= &\, 80h\alpha +hr^2(3+158h\gamma ))).\\ t_{17}(r)= &\, -102\gamma +r^2(-3+3hr^2(-1+hr^2+h^2r^4)-2h(129+hr^2(89+43hr^2))\gamma +r^{2\alpha }t_1(r)^3\alpha ^2). \end{aligned}$$