Anisotropic stellar Finch-Skea structures satisfying Karmarkar condition in a teleparallel framework involving off-diagonal tetrad

Abstract

The present paper is devoted to study the spherically symmetric compact star model with anisotropic matter distribution in the framework of f(T) modified gravity. In this work, we choose off-diagonal tetrad along with the power law modal of f(T), where T is the torsion scalar. For the sake of simplicity in calculations, we use Karmarkar condition and develop a significant differential equation providing the connection between two key elements $e^a$ and $e^b$ of the spacetime metric. Further, in order to determine the exact stellar solutions and values of involved constants, we take a well-known Finch-Skea structure into account as $g_{rr}$ component and find the resulting form of $g_{tt}$ component. In order to check the behavior of anisotropic fluid and to explore the stability of compact star, we use observed values of mass and radius for three compact stars: $PSR J1614-2230$, $Vela X-I$ and $SAX J1808.4-3658$. From the graphical illustration of different physical features of this model, it is concluded that our constructed stellar structure is physically viable and interesting.

Appendix

$$\begin{aligned}&f_1(r)\\&\quad =\sqrt{c r^2+d^{m-1} r^m+1},\\&f_2(r)\\&\quad =\sqrt{c r^2+d^{m-1} r^m},\\&f_3(r)\\&\quad =(m-6) \left( A d (m+2) \sqrt{c r^2+d^{m-1} r^m}+2 B r \left( c d r^2+d^m r^m\right) \right) \\&\qquad -2 B c (m-2) d^{1-m} r^{3-m} \left( c d r^2+d^m r^m\right) \\&\qquad \times \, _2F_1\left( 1,\frac{m-4}{m-2};\frac{3}{2}-\frac{2}{m-2};-c d^{1-m} r^{2-m}\right) ,\\&f_4(r)\\&\quad =\frac{d \left( f_1(r)-1\right) }{r^2 \left( c d r^2+d^m r^m+d\right) } \left( \frac{-2B(m-6)(m+2)r\left( c d r^2+d^m r^m\right) +f_1(r)-1}{f_3(r)}\right) ,\\&f_5(r)\\&\quad =2 Bc(m-3)(m-2)d^{1-m} r^{2-m} \left( c d r^2+d^m r^m\right) \, _2F_1\left( 1,\frac{m-4}{m-2};\frac{3}{2}-\frac{2}{m-2};-c d^{1-m} r^{2-m}\right) ,\\&f_6(r)\\&\quad = 2 Bc(m-2)d^{1-m} r^{2-m}\left( 2 c d r^2+m d^m r^m\right) \, _2F_1\left( 1,\frac{m-4}{m-2};\frac{3}{2}-\frac{2}{m-2};-c d^{1-m} r^{2-m}\right) ,\\&f_7(r)\\&\quad =\frac{4 B c^2 (m-4) (m-2)^2 d^{2-2 m} r^{4-2 m} \left( c d r^2+d^m r^m\right) \, _2F_1\left( 2,\frac{2 (m-3)}{m-2};\frac{5}{2}-\frac{2}{m-2};-c d^{1-m} r^{2-m}\right) }{3 m-10},\\&f_8(r) \\&\quad = \frac{2 B (m-6) (m+2) \left( c d r^2+d^m r^m\right) }{f_3(r)}, \end{aligned}$$

Also,

$$\begin{aligned}&g(r)\\&\quad = R^3(-(2 M-R))\left( c^2 d^2 R^4+c d R^2\left( 2 d^m R^m+d\right) +d^m R^m \left( d^m R^m\right. \right. \\&\qquad \left. \left. +d\right) \right) \Big [d^2 \Big [R^4 \left( 16 \beta ^2 c^2+\beta c \left( 56-32 f_1(r)\right) +1\right) \\&\qquad - 128 \beta ^2 \left( f_1(r)-1\right) +16 \beta R^2 \left( \beta c \left( 8-4 f_1(r)\right) -3 f_1(r)+3\right) +c R^6 (8 \beta c+1)\Big ]\\&\qquad +d^{m+1} R^m \Big [-64 \beta ^2 \left( f_1(r)-2\right) \\&\qquad + 8\beta R^2 \left( 4 \beta c-4 f_1(r)+7\right) +R^4 (16 \beta c+1)\Big ] \\&\qquad +8 \beta d^{2 m} R^{2 m} \left( 2 \beta +R^2\right) \Big ]. \end{aligned}$$