Physically viable solutions of anisotropic spheres in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="script">R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> gravity satisfying the Karmarkar condition

This paper is devoted to discussing compact stars in $f(\mathcal{R},\mathcal{G})$ gravity, where $\mathcal{R}$ and $\mathcal{G}$ denote the Ricci scalar and Gauss-Bonnet invariant, respectively. To meet this aim, we consider spherically symmetric space-time with an anisotropic fluid distribution. In particular, the Karmarkar condition is used to explore the compact star solutions. Furthermore, we choose two specific models of compact stars, namely LMC X-4 ($\mathrm{mass}=1.29M/{M}_{\ensuremath{\bigodot}}$ and $\mathrm{radii}=9.711\text{ }\text{ }\mathrm{km}$) and EXO 1785-248 ($\mathrm{mass}=1.30M/{M}_{\ensuremath{\bigodot}}$ and $\mathrm{radii}=8.849\text{ }\text{ }\mathrm{km}$). We develop the field equations for $f(\mathcal{R},\mathcal{G})$ gravity by employing the Karmarkar condition with a specific model already reported in literature by Lake [Phys. Rev. D 67, 104015 (2003)]. We further consider the Schwarzschild geometry for the matching conditions at the boundary. It is important to mention here that we calculate the values of all the involved parameters by imposing the matching condition. We have provided a detailed graphical analysis to discuss the physical acceptability of parameters, i.e., energy density, pressure, anisotropy, and gradients. We have also examined the stability of compacts stars by exploring the energy conditions, equation of state, generalized Tolman-Oppenheimer-Volkoff equation, causality condition, and adiabatic index. For the present analysis, we predict some numerical values in tabular form for central gravitational metric functions, central density, and central pressures components. We have also calculated the ratio ${p}_{rc}/{\ensuremath{\rho}}_{c}$ to check the validity of Zeldovich's condition. Conclusively, it is found that our obtained solutions are physically viable with a well-behaved nature in $f(\mathcal{R},\mathcal{G})$ modified gravity for the compact star models under discussion.

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